\[\operatorname{Ln}\Gamma\left(z\right)=\left(z-\tfrac{1}{2}\right)\ln z-z+%
\tfrac{1}{2}\ln\left(2\pi\right)+\int_{0}^{\infty}{\mathrm{e}}^{-zt}\left(%
\frac{1}{{\mathrm{e}}^{t}-1}-\frac{1}{t}+\frac{1}{2}\right)\frac{\,\mathrm{d}t%
}{t},\]