Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$.

What is its $\zeta$-regularized determinant?

This should be well-known, I suppose, but I didn't find a reference.

Some background: The eigenvalues of $L$ are $n^2+c$ for $n \in \mathbb{Z}$ (with multiplicity two each), and therefore the zeta-function for positive $c$ is given by $$\zeta_c(s) = 2\sum_{n=0}^\infty \frac{1}{(n^2+c)^s}.$$ Now the $\zeta$-regularized determinant is defined by $$\det(L) = e^{-\zeta^\prime(0)}.$$ How does one compute such a thing?

Consider on the circle $S^1$ the operator $$L := - \frac{\partial^2}{\partial \theta^2} + c$$ for some constant $c \in \mathbb{R}$.

What is its $\zeta$-regularized determinant?

This should be well-known, I suppose, but I didn't find a reference.

Some background: The eigenvalues of $L$ are $n^2+c$ for $n \in \mathbb{Z}$ (with multiplicity two each), and therefore the zeta-function for positive $c$ is given by $$\zeta_c(s) = 2\sum_{n=0}^\infty \frac{1}{(n^2+c)^s}.$$ Now the $\zeta$-regularized determinant is defined by $$\det(L) = e^{-\zeta^\prime(0)}.$$ How does one compute such a thing?

\Edit: The hint to look for "thermal zeta functions" was good: Here is a paper that computes the determinant in question: http://arxiv.org/pdf/hep-th/9505154v1.pdf