Reading a paper about eta invariants I came across a zeta-like function.

I'm looking for the analytic continuation of $$\sum_{k=1}^\infty k(k+a)^{-s}$$ at $s=0$, where $a$ is positive.

In the paper he just says "The [...] term causes no problem and at $s=0$ has the value $\left[\frac{(4a^2-1)}{12}\right]$." Unfortunately, I really don't see the that.

My first approach so far; I tried a Taylor series at $s=3$: \begin{align} \sum_{k=1}^\infty k(k+a)^{-s} &= \sum_{k=1}^\infty k \sum_{l=0}^\infty (-1)^l \begin{pmatrix}2+l \\ l\end{pmatrix} (k+a)^{-3-l} (s-3)^{l} \end{align} and inserted $s=0$ $$\sum_{k=1}^\infty \frac{k}{(k+a)^3} \sum_{l=0}^\infty \begin{pmatrix}2+l \\ l\end{pmatrix} \left( \frac{3}{k+a} \right)^l$$ which is for $a\geq 3$ $$ \sum_{k=1}^\infty \frac{k}{(k+a-3)^3}~. $$ That seems to converge if I didn't miscalculate. But what are the further steps in order to get the result above? Or is there a more skilful approach?

Thanks in advance!

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