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studying the article "On the Number of Prime Numbers less than a Given Quantity" I have noticed that Riemann derives the equation:

$$2\sin(\pi s)\Gamma(s)\zeta(s)= i\int_{C}\frac{(-x)^{s-1}}{e^x-1}dx$$

where the integral is from $+\infty$ to $+\infty$ circling the origin, and from this equation claims that $\zeta(s)=0$ for any $s$ negative even integer, I know this is a known property of the Zeta function and similar questions have already been asked, but none of the answers use this particular equation to derive this property, does someone know how to?

Zackury
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    This may give you some idea: https://math.stackexchange.com/questions/143449/riemanns-thinking-on-symmetrizing-the-zeta-functional-equation?rq=1 –  Oct 13 '24 at 11:14
  • Hi, I saw the post, It doesn't really explain how to get to the formula $$\zeta(-m)=\frac{(-1)^m}{2\pi i}\int_{|z|=1}\frac{m!}{z^{m+1}}\cdot\frac{1}{e^z-1}dz$$ – Zackury Oct 14 '24 at 08:10

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