\[H\left(s\right)=\frac{1}{2}\zeta\left(s+1\right)+\frac{\zeta\left(s\right)}{s-%
1}-\sum_{r=1}^{k}\genfrac{(}{)}{0.0pt}{}{s+2r-2}{2r-1}\zeta\left(1-2r\right)%
\zeta\left(s+2r\right)-\genfrac{(}{)}{0.0pt}{}{s+2k}{2k+1}\sum_{n=1}^{\infty}%
\frac{1}{n}\int_{n}^{\infty}\frac{\widetilde{B}_{2k+1}\left(x\right)}{x^{s+2k+%
1}}\,\mathrm{d}x.\]