\[\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\frac{1}{\Gamma\left(s\right)}%
\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x%
^{s-1}}{e^{x}}\,\mathrm{d}x,\]