Based on this question, Conjecture $\Re\,\operatorname{Li}_2\left(\frac12+\frac i6\right)=\frac{7\pi^2}{48}-\frac13\arctan^22-\frac16\arctan^23-\frac18\ln^2(\tfrac{18}5)$
We have this identity, which can be proved using the reflection formula or integration:
$$ \Re{{\rm Li}_{2}\left(\frac{1}{2}+iq\right)}=\frac{{\pi}^{2}}{12}-\frac{1}{8}{\ln{\left(\frac{1+4q^2}{4}\right)}}^{2}-\frac{{\arctan{(2q)}}^{2}}{2} $$
Are there any others like this without the real part being restricted to 1/2? Obviously if the complex part is zero, then we have the usual ones like 1/2,2,-1, $\phi$, etc.
