# Formula:DLMF:25.11:E30

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

## Constraint(s)

${\displaystyle{\displaystyle{\displaystyle s\neq 1}}}$ &
${\displaystyle{\displaystyle{\displaystyle\realpart{a}>0}}}$ &

the integration contour is a loop around the negative real axis; it starts at ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$, encircles the origin once in the positive direction without enclosing any of the points

${\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi i}}}$, ${\displaystyle{\displaystyle{\displaystyle\pm 4\pi i,\ldots,}}}$ and returns to ${\displaystyle{\displaystyle{\displaystyle-\infty}}}$

## Proof

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