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