Value of polygamma at 1

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Theorem

The following formula holds for $m=1,2,3,\ldots$: $$\psi^{(m)}(1)=(-1)^{m+1} m! \zeta(m+1),$$ where $\psi^{(m)}$ denotes the polygamma, $m!$ denotes the factorial$, and $\zeta$ denotes the Riemann zeta function.

Proof

Reference