What is the name of this zeta function? I'm looking for literature – in particular, methods of evaluation – for zeta functions defined by the sum
$$\zeta(s,a) = \sum_{k=-\infty}^\infty \frac{1}{(k^2+a)^s}\,.$$
I am interested in the continuation to $s=-3/2$.
It looks like the Hurwitz zeta function, except that in the denominator, we have $k^2$ instead of $k$.  Is there any work on this kind of sum?  Does this zeta function have a name?
 A: I don't have any literature for you, and I have no insight about the case of $s=-3/2$, but I can provide some methods for evaluation. First of all, I'd like you to recall the formula
$$\zeta(1,a)=\sum_{k=-\infty}^\infty \frac{1}{k^2+a}=\frac{\pi \coth(\pi \sqrt a)}{\sqrt a}$$
This formula can be easily derived using the residue theorem, which I am just assuming that you already know how to use. If not, correct me and I will write you a short proof as soon as I can.
Now observe the following interesting and useful property of the function $\zeta$ as you have defined it. If $s\gt 0$, then
$$\begin{align}
\frac{d}{da}\zeta(s,a)
&= \frac{d}{da} \sum_{k=-\infty}^\infty \frac{1}{(k^2+a)^s}\\
&= \sum_{k=-\infty}^\infty \frac{d}{da} \frac{1}{(k^2+a)^s}\\
&= \sum_{k=-\infty}^\infty \frac{-s}{(k^2+a)^{s+1}}\\
\end{align}$$
and so we have the following recurrence:
$$\zeta(s+1,a)=-\frac{1}{s}\frac{d}{da}\zeta(s,a),\space\space\space s\gt 0$$
From this recurrence, we can derive the following formulas:
$$\zeta(2,a)=\frac{\pi\coth(\pi\sqrt a)}{2a^{3/2}}+\frac{\pi^2\text{csch}^2(\pi\sqrt a)}{2a}$$
$$\zeta(3,a)=\frac{3\pi\coth(\pi \sqrt a)}{8a^{5/2}}+\frac{\pi^3\coth(\pi\sqrt a)\text{csch}^2(\pi\sqrt a)}{4a^{3/2}}+\frac{3\pi^2\text{csch}^2(\pi\sqrt a)}{8a^2}$$
I could keep going, but the algebra just keeps getting messier. So instead, I will generalize this by writing
$$\color{green}{\zeta(n,a)=\frac{(-1)^{n+1}}{(n-1)!}\frac{d^{n-1}}{dx^{n-1}}\frac{\pi \coth(\pi \sqrt x)}{\sqrt x}_{x=a}}$$
which holds for all $n\in\mathbb N$ and $a$ not a negative perfect square.
