# Formula:DLMF:25.5:E21

${\displaystyle{\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma% \left(1-s\right)}{2\cpi\iunit(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}% {\expe^{-z}+1}\diffd z}}}$

## Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}}$ &

The contour here is any loop that encircles the origin in the positive direction

not enclosing any of the points ${\displaystyle{\displaystyle{\displaystyle\pm\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 3\pi i,\ldots}}}$

## Proof

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