I have the following series:
$$ \sum_{n = 0}^{+\infty} \frac{n^2}{(n^2 + a^2)^{\epsilon}} $$
with $a\in \mathbb{R}$. How can I find its analytic continuation for $\epsilon \in \mathbb{C}$? In particular the case $\epsilon = -1/2$ is what I need to find. I tried proceeding in the same way as in the case of the analytic continuation of the zeta function, but i'm stuck with a sum over Gaussian functions. I'm wondering if it can be written as a particular case of the Epstein zeta function.