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