\[\zeta'\left(s,a\right)=-\frac{\ln a}{a^{s}}\left(\frac{1}{2}+\frac{a}{s-1}%
\right)-\frac{a^{1-s}}{(s-1)^{2}}+\frac{s(s+1)}{2}\int_{0}^{\infty}\frac{(%
\widetilde{B}_{2}\left(x\right)-B_{2})\ln\left(x+a\right)}{(x+a)^{s+2}}\,%
\mathrm{d}x-\frac{(2s+1)}{2}\int_{0}^{\infty}\frac{\widetilde{B}_{2}\left(x%
\right)-B_{2}}{(x+a)^{s+2}}\,\mathrm{d}x,\]