We know that some expressions of $\zeta (s)$ contains the factor: ${\frac {1}{e^{x}-1}}$.
For example:
${\displaystyle 2\sin(\pi s)\Gamma (s)\zeta (s)=i\oint _{H}{\frac {(-x)^{s-1}} {e^{x}-1}}\,\mathrm {d} x}$
For Riemann $\xi (s)$:
$\xi (s)={\tfrac {1}{2}}s(s-1)\pi ^{{-s/2}}\Gamma \left({\tfrac {1}{2}}s\right)\zeta (s)$
Is there a simple expression for it which contains the factor ${\frac {1}{e^{x}-1}}$ ?
In other words, is there a simple relation between $\xi (s)$ and the factor ${\frac {1}{e^{x}-1}}$ ?