I am interested in a certain "zeta function," a meromorphic function of $s \in \mathbb{C}$ that depends on a real parameter $\alpha \neq 0$. It's defined for the real part of $s$ large by $$ \zeta_\alpha(s) = \sum_{k \in \mathbb{Z}} (k^2 + \alpha^2)^{-s} $$ I think it has a meromorphic extension that is analytic at $s=0$. My question is:
What is $\zeta_\alpha'(0)$?
My interest in this question is that $\zeta_\alpha'(0)$ is the determinant of the operator $-\frac{d^2}{dx^2} + \alpha^2$ on the circle, but I think the question makes independent of this context.
My apologies, because I'm sure the answer is well-known, so a pointer to a reference would be great (in fact, I'm sure I've seen it somewhere, but I can't find it now).
Edit: As I was writing this question, I found the Wikipedia article on functional determinants, which discusses this problem. But I thought I would ask the question anyway, to see if anyone has a better reference, or any other insight into this question.
I think the best place to look is the book "Zeta Regularization Techniques with Applications" by Eliazade et. al. There are some other books by Eliazade, which I think you might find useful.
The calculation you want is fairly straightforward. If it will help, I will sketch how to do it later.