It is not too hard to prove:
$$
\int_{0}^{\infty} \left(\frac{1}{z^5(e^z-1)}+\frac{1}{720z^2} -\frac{1}{12z^4}
+\frac{1}{2z^5}-\frac{1}{z^6} \right )\text{d}z=\frac{\zeta(5)}{32\pi^4}
$$
And
$$
\int_{0}^{\infty} \left(\frac{1}{z^7(e^z-1)}
-\frac{1}{30240z^2} +\frac{1}{720z^4} -\frac{1}{12z^6}
+\frac{1}{2z^7}-\frac{1}{z^8} \right )\text{d}z=-\frac{\zeta(7)}{128\pi^6}
$$
Now I am considering the Dirichlet's L-series:
$$
L_k(s,\chi)=\sum_{n=1}^{\infty} \frac{\chi_k(n)}{n^s}
$$
I also found
$$
\int_{-\infty}^{\infty} \left ( \frac{1}{x^{10}(2\cosh x+1)}
-\frac{809}{544320x^2}+\frac{7}{1080x^4}-\frac{1}{36x^6}+\frac{1}{9x^8}-\frac{1}{3x^{10}}
\right ) \text{d}x
=\frac{671\pi}{544320\sqrt{3} } -\frac{\psi^{(9)}\left ( \frac{1}{3} \right ) }{92897280\sqrt{3}\pi^9 }
$$
My question is:
Are there any generalized forms about the integrals?
The equivalent problem is to solve
$$
\lim_{z \to -n} \Gamma(z)L_k(z,\chi),
$$
where $n$ is a positive integer.
- 5,024
- 6,789
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I think I had the same question with some generalization (set $a=1$, $x\to 0$). https://math.stackexchange.com/q/3953436/193034 – tyobrien Jul 24 '21 at 14:26
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It would be better to phrase your question more clearly, in its current state, it is literally incomprehensible (both flow of logic and unnecessarily complicated selected examples). Regarding your question (WLOG assume $\chi$ is primitive) $\lim_{z \to -n} \Gamma(z)L(z,\chi)$, this limit exists iff $\chi$ has same parity as $n$, in this case, it equals an algebraic multiple of $\pi^{-n}$ and $L(n+1,\overline{\chi})$. This $L(n+1,\overline{\chi})$ has no simpler form (or like what you did in your last example, express it as polygamma). – pisco Jul 26 '21 at 09:45
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1You say « it’s not too hard to prove… » Could you be kind enough to give more details on how you solve your proposed integral – Fjaclot Jul 26 '21 at 13:32
1 Answers
Here'a a table of results generated by Mathematica which can be used to investigate patterns.
$\begin{array}{cccc} k & j & n & \text{Limit[$\Gamma[z]\ L_{k,j}(z), z\to n$]} \\ 1 & 1 & -10 & -\frac{\zeta (11)}{2048 \pi ^{10}} \\ 1 & 1 & -9 & \text{Indeterminate} \\ 1 & 1 & -8 & \frac{\zeta (9)}{512 \pi ^8} \\ 1 & 1 & -7 & \text{Indeterminate} \\ 1 & 1 & -6 & -\frac{\zeta (7)}{128 \pi ^6} \\ 1 & 1 & -5 & \text{Indeterminate} \\ 1 & 1 & -4 & \frac{\zeta (5)}{32 \pi ^4} \\ 1 & 1 & -3 & \text{Indeterminate} \\ 1 & 1 & -2 & -\frac{\zeta (3)}{8 \pi ^2} \\ 1 & 1 & -1 & \text{Indeterminate} \\ 1 & 1 & 0 & \text{Indeterminate} \\ 1 & 1 & 1 & \text{Indeterminate} \\ 1 & 1 & 2 & \frac{\pi ^2}{6} \\ 1 & 1 & 3 & 2 \zeta (3) \\ 1 & 1 & 4 & \frac{\pi ^4}{15} \\ 1 & 1 & 5 & 24 \zeta (5) \\ 1 & 1 & 6 & \frac{8 \pi ^6}{63} \\ 1 & 1 & 7 & 720 \zeta (7) \\ 1 & 1 & 8 & \frac{8 \pi ^8}{15} \\ 1 & 1 & 9 & 40320 \zeta (9) \\ 1 & 1 & 10 & \frac{128 \pi ^{10}}{33} \\ \text{} & \text{} & \text{} & \text{} \\ 2 & 1 & -10 & \frac{1023 \zeta (11)}{2048 \pi ^{10}} \\ 2 & 1 & -9 & \text{Indeterminate} \\ 2 & 1 & -8 & -\frac{255 \zeta (9)}{512 \pi ^8} \\ 2 & 1 & -7 & \text{Indeterminate} \\ 2 & 1 & -6 & \frac{63 \zeta (7)}{128 \pi ^6} \\ 2 & 1 & -5 & \text{Indeterminate} \\ 2 & 1 & -4 & -\frac{15 \zeta (5)}{32 \pi ^4} \\ 2 & 1 & -3 & \text{Indeterminate} \\ 2 & 1 & -2 & \frac{3 \zeta (3)}{8 \pi ^2} \\ 2 & 1 & -1 & \text{Indeterminate} \\ 2 & 1 & 0 & -\frac{\log (2)}{2} \\ 2 & 1 & 1 & \text{Indeterminate} \\ 2 & 1 & 2 & \frac{\pi ^2}{8} \\ 2 & 1 & 3 & \frac{7 \zeta (3)}{4} \\ 2 & 1 & 4 & \frac{\pi ^4}{16} \\ 2 & 1 & 5 & \frac{93 \zeta (5)}{4} \\ 2 & 1 & 6 & \frac{\pi ^6}{8} \\ 2 & 1 & 7 & \frac{5715 \zeta (7)}{8} \\ 2 & 1 & 8 & \frac{17 \pi ^8}{32} \\ 2 & 1 & 9 & \frac{160965 \zeta (9)}{4} \\ 2 & 1 & 10 & \frac{31 \pi ^{10}}{8} \\ \text{} & \text{} & \text{} & \text{} \\ 3 & 1 & -10 & \frac{7381 \zeta (11)}{256 \pi ^{10}} \\ 3 & 1 & -9 & \text{Indeterminate} \\ 3 & 1 & -8 & -\frac{205 \zeta (9)}{16 \pi ^8} \\ 3 & 1 & -7 & \text{Indeterminate} \\ 3 & 1 & -6 & \frac{91 \zeta (7)}{16 \pi ^6} \\ 3 & 1 & -5 & \text{Indeterminate} \\ 3 & 1 & -4 & -\frac{5 \zeta (5)}{2 \pi ^4} \\ 3 & 1 & -3 & \text{Indeterminate} \\ 3 & 1 & -2 & \frac{\zeta (3)}{\pi ^2} \\ 3 & 1 & -1 & \text{Indeterminate} \\ 3 & 1 & 0 & -\frac{\log (3)}{2} \\ 3 & 1 & 1 & \text{Indeterminate} \\ 3 & 1 & 2 & \frac{4 \pi ^2}{27} \\ 3 & 1 & 3 & \frac{52 \zeta (3)}{27} \\ 3 & 1 & 4 & \frac{16 \pi ^4}{243} \\ 3 & 1 & 5 & \frac{1936 \zeta (5)}{81} \\ 3 & 1 & 6 & \frac{832 \pi ^6}{6561} \\ 3 & 1 & 7 & \frac{174880 \zeta (7)}{243} \\ 3 & 1 & 8 & \frac{10496 \pi ^8}{19683} \\ 3 & 1 & 9 & \frac{88175360 \zeta (9)}{2187} \\ 3 & 1 & 10 & \frac{687104 \pi ^{10}}{177147} \\ \text{} & \text{} & \text{} & \text{} \\ 3 & 2 & -10 & \text{Indeterminate} \\ 3 & 2 & -9 & -\frac{243 \left(\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{1}{3}\right)-\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{2}{3}\right)\right)}{4480} \\ 3 & 2 & -8 & \text{Indeterminate} \\ 3 & 2 & -7 & \frac{1}{560} (-243) \left(\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{1}{3}\right)-\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{2}{3}\right)\right) \\ 3 & 2 & -6 & \text{Indeterminate} \\ 3 & 2 & -5 & \frac{1}{40} (-81) \left(\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{1}{3}\right)-\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{2}{3}\right)\right) \\ 3 & 2 & -4 & \text{Indeterminate} \\ 3 & 2 & -3 & \frac{1}{2} (-9) \left(\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{1}{3}\right)-\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{2}{3}\right)\right) \\ 3 & 2 & -2 & \text{Indeterminate} \\ 3 & 2 & -1 & 3 \text{HurwitzZeta}^{(1,0)}\left(-1,\frac{2}{3}\right)-3 \text{HurwitzZeta}^{(1,0)}\left(-1,\frac{1}{3}\right) \\ 3 & 2 & 0 & \text{Indeterminate} \\ 3 & 2 & 1 & \frac{\pi }{3 \sqrt{3}} \\ 3 & 2 & 2 & \frac{1}{9} \left(\psi ^{(1)}\left(\frac{1}{3}\right)-\psi ^{(1)}\left(\frac{2}{3}\right)\right) \\ 3 & 2 & 3 & \frac{8 \pi ^3}{81 \sqrt{3}} \\ 3 & 2 & 4 & \frac{2}{27} \left(\zeta \left(4,\frac{1}{3}\right)-\zeta \left(4,\frac{2}{3}\right)\right) \\ 3 & 2 & 5 & \frac{32 \pi ^5}{243 \sqrt{3}} \\ 3 & 2 & 6 & \frac{1}{729} \left(\psi ^{(5)}\left(\frac{1}{3}\right)-\psi ^{(5)}\left(\frac{2}{3}\right)\right) \\ 3 & 2 & 7 & \frac{896 \pi ^7}{2187 \sqrt{3}} \\ 3 & 2 & 8 & \frac{\psi ^{(7)}\left(\frac{1}{3}\right)-\psi ^{(7)}\left(\frac{2}{3}\right)}{6561} \\ 3 & 2 & 9 & \frac{414208 \pi ^9}{177147 \sqrt{3}} \\ 3 & 2 & 10 & \frac{\psi ^{(9)}\left(\frac{1}{3}\right)-\psi ^{(9)}\left(\frac{2}{3}\right)}{59049} \\ \text{} & \text{} & \text{} & \text{} \\ 4 & 1 & -10 & \frac{1023 \zeta (11)}{2048 \pi ^{10}} \\ 4 & 1 & -9 & \text{Indeterminate} \\ 4 & 1 & -8 & -\frac{255 \zeta (9)}{512 \pi ^8} \\ 4 & 1 & -7 & \text{Indeterminate} \\ 4 & 1 & -6 & \frac{63 \zeta (7)}{128 \pi ^6} \\ 4 & 1 & -5 & \text{Indeterminate} \\ 4 & 1 & -4 & -\frac{15 \zeta (5)}{32 \pi ^4} \\ 4 & 1 & -3 & \text{Indeterminate} \\ 4 & 1 & -2 & \frac{3 \zeta (3)}{8 \pi ^2} \\ 4 & 1 & -1 & \text{Indeterminate} \\ 4 & 1 & 0 & -\frac{\log (2)}{2} \\ 4 & 1 & 1 & \text{Indeterminate} \\ 4 & 1 & 2 & \frac{\pi ^2}{8} \\ 4 & 1 & 3 & \frac{7 \zeta (3)}{4} \\ 4 & 1 & 4 & \frac{\pi ^4}{16} \\ 4 & 1 & 5 & \frac{93 \zeta (5)}{4} \\ 4 & 1 & 6 & \frac{\pi ^6}{8} \\ 4 & 1 & 7 & \frac{5715 \zeta (7)}{8} \\ 4 & 1 & 8 & \frac{17 \pi ^8}{32} \\ 4 & 1 & 9 & \frac{160965 \zeta (9)}{4} \\ 4 & 1 & 10 & \frac{31 \pi ^{10}}{8} \\ \text{} & \text{} & \text{} & \text{} \\ 4 & 2 & -10 & \text{Indeterminate} \\ 4 & 2 & -9 & \frac{31 \pi }{5670}-\frac{\psi ^{(9)}\left(\frac{1}{4}\right)}{371589120 \pi ^9} \\ 4 & 2 & -8 & \text{Indeterminate} \\ 4 & 2 & -7 & \frac{\psi ^{(7)}\left(\frac{1}{4}\right)-17408 \pi ^8}{1290240 \pi ^7} \\ 4 & 2 & -6 & \text{Indeterminate} \\ 4 & 2 & -5 & \frac{\pi }{30}-\frac{\psi ^{(5)}\left(\frac{1}{4}\right)}{7680 \pi ^5} \\ 4 & 2 & -4 & \text{Indeterminate} \\ 4 & 2 & -3 & \frac{\psi ^{(3)}\left(\frac{1}{4}\right)-8 \pi ^4}{96 \pi ^3} \\ 4 & 2 & -2 & \text{Indeterminate} \\ 4 & 2 & -1 & -\frac{2 C}{\pi } \\ 4 & 2 & 0 & \text{Indeterminate} \\ 4 & 2 & 1 & \frac{\pi }{4} \\ 4 & 2 & 2 & C \\ 4 & 2 & 3 & \frac{\pi ^3}{16} \\ 4 & 2 & 4 & \frac{3}{128} \left(\zeta \left(4,\frac{1}{4}\right)-\zeta \left(4,\frac{3}{4}\right)\right) \\ 4 & 2 & 5 & \frac{5 \pi ^5}{64} \\ 4 & 2 & 6 & \frac{15}{512} \left(\zeta \left(6,\frac{1}{4}\right)-\zeta \left(6,\frac{3}{4}\right)\right) \\ 4 & 2 & 7 & \frac{61 \pi ^7}{256} \\ 4 & 2 & 8 & \frac{\psi ^{(7)}\left(\frac{1}{4}\right)-\psi ^{(7)}\left(\frac{3}{4}\right)}{65536} \\ 4 & 2 & 9 & \frac{1385 \pi ^9}{1024} \\ 4 & 2 & 10 & \frac{\psi ^{(9)}\left(\frac{1}{4}\right)-\psi ^{(9)}\left(\frac{3}{4}\right)}{1048576} \\ \text{} & \text{} & \text{} & \text{} \\ 5 & 1 & -10 & \frac{1220703 \zeta (11)}{256 \pi ^{10}} \\ 5 & 1 & -9 & \text{Indeterminate} \\ 5 & 1 & -8 & -\frac{12207 \zeta (9)}{16 \pi ^8} \\ 5 & 1 & -7 & \text{Indeterminate} \\ 5 & 1 & -6 & \frac{1953 \zeta (7)}{16 \pi ^6} \\ 5 & 1 & -5 & \text{Indeterminate} \\ 5 & 1 & -4 & -\frac{39 \zeta (5)}{2 \pi ^4} \\ 5 & 1 & -3 & \text{Indeterminate} \\ 5 & 1 & -2 & \frac{3 \zeta (3)}{\pi ^2} \\ 5 & 1 & -1 & \text{Indeterminate} \\ 5 & 1 & 0 & -\frac{\log (5)}{2} \\ 5 & 1 & 1 & \text{Indeterminate} \\ 5 & 1 & 2 & \frac{4 \pi ^2}{25} \\ 5 & 1 & 3 & \frac{248 \zeta (3)}{125} \\ 5 & 1 & 4 & \frac{208 \pi ^4}{3125} \\ 5 & 1 & 5 & \frac{74976 \zeta (5)}{3125} \\ 5 & 1 & 6 & \frac{1984 \pi ^6}{15625} \\ 5 & 1 & 7 & \frac{11249856 \zeta (7)}{15625} \\ 5 & 1 & 8 & \frac{1041664 \pi ^8}{1953125} \\ 5 & 1 & 9 & \frac{15749991936 \zeta (9)}{390625} \\ 5 & 1 & 10 & \frac{37878784 \pi ^{10}}{9765625} \\ \text{} & \text{} & \text{} & \text{} \\ 5 & 2 & -10 & \text{Indeterminate} \\ 5 & 2 & -9 & \frac{390625 \left(-\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{1}{5}\right)-i \text{HurwitzZeta}^{(1,0)}\left(-9,\frac{2}{5}\right)+i \text{HurwitzZeta}^{(1,0)}\left(-9,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{4}{5}\right)\right)}{72576} \\ 5 & 2 & -8 & \text{Indeterminate} \\ 5 & 2 & -7 & \frac{15625 \left(-\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{1}{5}\right)-i \text{HurwitzZeta}^{(1,0)}\left(-7,\frac{2}{5}\right)+i \text{HurwitzZeta}^{(1,0)}\left(-7,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{4}{5}\right)\right)}{1008} \\ 5 & 2 & -6 & \text{Indeterminate} \\ 5 & 2 & -5 & \frac{625}{24} \left(-\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{1}{5}\right)-i \text{HurwitzZeta}^{(1,0)}\left(-5,\frac{2}{5}\right)+i \text{HurwitzZeta}^{(1,0)}\left(-5,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{4}{5}\right)\right) \\ 5 & 2 & -4 & \text{Indeterminate} \\ 5 & 2 & -3 & \frac{125}{6} \left(-\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{1}{5}\right)-i \text{HurwitzZeta}^{(1,0)}\left(-3,\frac{2}{5}\right)+i \text{HurwitzZeta}^{(1,0)}\left(-3,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{4}{5}\right)\right) \\ 5 & 2 & -2 & \text{Indeterminate} \\ 5 & 2 & -1 & 5 \left(-\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{1}{5}\right)-i \text{HurwitzZeta}^{(1,0)}\left(-1,\frac{2}{5}\right)+i \text{HurwitzZeta}^{(1,0)}\left(-1,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{4}{5}\right)\right) \\ 5 & 2 & 0 & \text{Indeterminate} \\ 5 & 2 & 1 & \frac{1}{5} \left(-\psi ^{(0)}\left(\frac{1}{5}\right)+i \left(\psi ^{(0)}\left(\frac{3}{5}\right)-\psi ^{(0)}\left(\frac{2}{5}\right)\right)+\psi ^{(0)}\left(\frac{4}{5}\right)\right) \\ 5 & 2 & 2 & \frac{1}{25} \left(\psi ^{(1)}\left(\frac{1}{5}\right)+i \left(\psi ^{(1)}\left(\frac{2}{5}\right)-\psi ^{(1)}\left(\frac{3}{5}\right)\right)-\psi ^{(1)}\left(\frac{4}{5}\right)\right) \\ 5 & 2 & 3 & \frac{2}{125} \left(\zeta \left(3,\frac{1}{5}\right)+i \left(\zeta \left(3,\frac{2}{5}\right)-\zeta \left(3,\frac{3}{5}\right)+i \zeta \left(3,\frac{4}{5}\right)\right)\right) \\ 5 & 2 & 4 & \frac{6}{625} \left(\zeta \left(4,\frac{1}{5}\right)+i \left(\zeta \left(4,\frac{2}{5}\right)-\zeta \left(4,\frac{3}{5}\right)+i \zeta \left(4,\frac{4}{5}\right)\right)\right) \\ 5 & 2 & 5 & \frac{24 \left(\zeta \left(5,\frac{1}{5}\right)+i \left(\zeta \left(5,\frac{2}{5}\right)-\zeta \left(5,\frac{3}{5}\right)+i \zeta \left(5,\frac{4}{5}\right)\right)\right)}{3125} \\ 5 & 2 & 6 & \frac{24 \left(\zeta \left(6,\frac{1}{5}\right)+i \left(\zeta \left(6,\frac{2}{5}\right)-\zeta \left(6,\frac{3}{5}\right)+i \zeta \left(6,\frac{4}{5}\right)\right)\right)}{3125} \\ 5 & 2 & 7 & \frac{144 \left(\zeta \left(7,\frac{1}{5}\right)+i \left(\zeta \left(7,\frac{2}{5}\right)-\zeta \left(7,\frac{3}{5}\right)+i \zeta \left(7,\frac{4}{5}\right)\right)\right)}{15625} \\ 5 & 2 & 8 & \frac{\psi ^{(7)}\left(\frac{1}{5}\right)+i \left(\psi ^{(7)}\left(\frac{2}{5}\right)-\psi ^{(7)}\left(\frac{3}{5}\right)+i \psi ^{(7)}\left(\frac{4}{5}\right)\right)}{390625} \\ 5 & 2 & 9 & \frac{8064 \left(\zeta \left(9,\frac{1}{5}\right)+i \left(\zeta \left(9,\frac{2}{5}\right)-\zeta \left(9,\frac{3}{5}\right)+i \zeta \left(9,\frac{4}{5}\right)\right)\right)}{390625} \\ 5 & 2 & 10 & \frac{\psi ^{(9)}\left(\frac{1}{5}\right)+i \left(\psi ^{(9)}\left(\frac{2}{5}\right)-\psi ^{(9)}\left(\frac{3}{5}\right)+i \psi ^{(9)}\left(\frac{4}{5}\right)\right)}{9765625} \\ \text{} & \text{} & \text{} & \text{} \\ 5 & 3 & -10 & \frac{390625 \left(\text{HurwitzZeta}^{(1,0)}\left(-10,\frac{1}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-10,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-10,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-10,\frac{4}{5}\right)\right)}{145152} \\ 5 & 3 & -9 & \text{Indeterminate} \\ 5 & 3 & -8 & \frac{78125 \left(\text{HurwitzZeta}^{(1,0)}\left(-8,\frac{1}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-8,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-8,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-8,\frac{4}{5}\right)\right)}{8064} \\ 5 & 3 & -7 & \text{Indeterminate} \\ 5 & 3 & -6 & \frac{3125}{144} \left(\text{HurwitzZeta}^{(1,0)}\left(-6,\frac{1}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-6,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-6,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-6,\frac{4}{5}\right)\right) \\ 5 & 3 & -5 & \text{Indeterminate} \\ 5 & 3 & -4 & \frac{625}{24} \left(\text{HurwitzZeta}^{(1,0)}\left(-4,\frac{1}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-4,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-4,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-4,\frac{4}{5}\right)\right) \\ 5 & 3 & -3 & \text{Indeterminate} \\ 5 & 3 & -2 & \frac{25}{2} \left(\text{HurwitzZeta}^{(1,0)}\left(-2,\frac{1}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-2,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-2,\frac{3}{5}\right)+\text{HurwitzZeta}^{(1,0)}\left(-2,\frac{4}{5}\right)\right) \\ 5 & 3 & -1 & \text{Indeterminate} \\ 5 & 3 & 0 & \text{log$\Gamma $}\left(-\frac{1}{5}\right)-\text{log$\Gamma $}\left(-\frac{2}{5}\right)-\text{log$\Gamma $}\left(-\frac{3}{5}\right)+\text{log$\Gamma $}\left(-\frac{4}{5}\right)-\log \left(\frac{3}{2}\right) \\ 5 & 3 & 1 & \frac{\log \left(\frac{1}{2} \left(\sqrt{5}+3\right)\right)}{\sqrt{5}} \\ 5 & 3 & 2 & \frac{4 \pi ^2}{25 \sqrt{5}} \\ 5 & 3 & 3 & \frac{2}{125} \left(\zeta \left(3,\frac{1}{5}\right)-\zeta \left(3,\frac{2}{5}\right)-\zeta \left(3,\frac{3}{5}\right)+\zeta \left(3,\frac{4}{5}\right)\right) \\ 5 & 3 & 4 & \frac{16 \pi ^4}{125 \sqrt{5}} \\ 5 & 3 & 5 & \frac{-\psi ^{(4)}\left(\frac{1}{5}\right)+\psi ^{(4)}\left(\frac{2}{5}\right)+\psi ^{(4)}\left(\frac{3}{5}\right)-\psi ^{(4)}\left(\frac{4}{5}\right)}{3125} \\ 5 & 3 & 6 & \frac{4288 \pi ^6}{15625 \sqrt{5}} \\ 5 & 3 & 7 & \frac{-\psi ^{(6)}\left(\frac{1}{5}\right)+\psi ^{(6)}\left(\frac{2}{5}\right)+\psi ^{(6)}\left(\frac{3}{5}\right)-\psi ^{(6)}\left(\frac{4}{5}\right)}{78125} \\ 5 & 3 & 8 & \frac{92416 \pi ^8}{78125 \sqrt{5}} \\ 5 & 3 & 9 & \frac{-\psi ^{(8)}\left(\frac{1}{5}\right)+\psi ^{(8)}\left(\frac{2}{5}\right)+\psi ^{(8)}\left(\frac{3}{5}\right)-\psi ^{(8)}\left(\frac{4}{5}\right)}{1953125} \\ 5 & 3 & 10 & \frac{422657024 \pi ^{10}}{48828125 \sqrt{5}} \\ \text{} & \text{} & \text{} & \text{} \\ 5 & 4 & -10 & \text{Indeterminate} \\ 5 & 4 & -9 & \frac{390625 \left(-\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{1}{5}\right)+i \left(\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{3}{5}\right)\right)+\text{HurwitzZeta}^{(1,0)}\left(-9,\frac{4}{5}\right)\right)}{72576} \\ 5 & 4 & -8 & \text{Indeterminate} \\ 5 & 4 & -7 & \frac{15625 \left(-\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{1}{5}\right)+i \left(\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{3}{5}\right)\right)+\text{HurwitzZeta}^{(1,0)}\left(-7,\frac{4}{5}\right)\right)}{1008} \\ 5 & 4 & -6 & \text{Indeterminate} \\ 5 & 4 & -5 & \frac{625}{24} \left(-\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{1}{5}\right)+i \left(\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{3}{5}\right)\right)+\text{HurwitzZeta}^{(1,0)}\left(-5,\frac{4}{5}\right)\right) \\ 5 & 4 & -4 & \text{Indeterminate} \\ 5 & 4 & -3 & \frac{125}{6} \left(-\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{1}{5}\right)+i \left(\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{3}{5}\right)\right)+\text{HurwitzZeta}^{(1,0)}\left(-3,\frac{4}{5}\right)\right) \\ 5 & 4 & -2 & \text{Indeterminate} \\ 5 & 4 & -1 & 5 \left(-\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{1}{5}\right)+i \left(\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{2}{5}\right)-\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{3}{5}\right)\right)+\text{HurwitzZeta}^{(1,0)}\left(-1,\frac{4}{5}\right)\right) \\ 5 & 4 & 0 & \text{Indeterminate} \\ 5 & 4 & 1 & \frac{1}{5} \left(-\psi ^{(0)}\left(\frac{1}{5}\right)+i \left(\psi ^{(0)}\left(\frac{2}{5}\right)-\psi ^{(0)}\left(\frac{3}{5}\right)\right)+\psi ^{(0)}\left(\frac{4}{5}\right)\right) \\ 5 & 4 & 2 & \frac{1}{25} \left(\psi ^{(1)}\left(\frac{1}{5}\right)+i \left(\psi ^{(1)}\left(\frac{3}{5}\right)-\psi ^{(1)}\left(\frac{2}{5}\right)\right)-\psi ^{(1)}\left(\frac{4}{5}\right)\right) \\ 5 & 4 & 3 & \frac{2}{125} \left(\zeta \left(3,\frac{1}{5}\right)-i \zeta \left(3,\frac{2}{5}\right)+i \zeta \left(3,\frac{3}{5}\right)-\zeta \left(3,\frac{4}{5}\right)\right) \\ 5 & 4 & 4 & \frac{6}{625} \left(\zeta \left(4,\frac{1}{5}\right)-i \zeta \left(4,\frac{2}{5}\right)+i \zeta \left(4,\frac{3}{5}\right)-\zeta \left(4,\frac{4}{5}\right)\right) \\ 5 & 4 & 5 & \frac{24 \left(\zeta \left(5,\frac{1}{5}\right)-i \zeta \left(5,\frac{2}{5}\right)+i \zeta \left(5,\frac{3}{5}\right)-\zeta \left(5,\frac{4}{5}\right)\right)}{3125} \\ 5 & 4 & 6 & \frac{24 \left(\zeta \left(6,\frac{1}{5}\right)-i \zeta \left(6,\frac{2}{5}\right)+i \zeta \left(6,\frac{3}{5}\right)-\zeta \left(6,\frac{4}{5}\right)\right)}{3125} \\ 5 & 4 & 7 & \frac{144 \left(\zeta \left(7,\frac{1}{5}\right)-i \zeta \left(7,\frac{2}{5}\right)+i \zeta \left(7,\frac{3}{5}\right)-\zeta \left(7,\frac{4}{5}\right)\right)}{15625} \\ 5 & 4 & 8 & \frac{\psi ^{(7)}\left(\frac{1}{5}\right)-i \psi ^{(7)}\left(\frac{2}{5}\right)+i \psi ^{(7)}\left(\frac{3}{5}\right)-\psi ^{(7)}\left(\frac{4}{5}\right)}{390625} \\ 5 & 4 & 9 & \frac{8064 \left(\zeta \left(9,\frac{1}{5}\right)-i \zeta \left(9,\frac{2}{5}\right)+i \zeta \left(9,\frac{3}{5}\right)-\zeta \left(9,\frac{4}{5}\right)\right)}{390625} \\ 5 & 4 & 10 & \frac{\psi ^{(9)}\left(\frac{1}{5}\right)-i \psi ^{(9)}\left(\frac{2}{5}\right)+i \psi ^{(9)}\left(\frac{3}{5}\right)-\psi ^{(9)}\left(\frac{4}{5}\right)}{9765625} \\ \text{} & \text{} & \text{} & \text{} \\ \end{array}$
- 8,744