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Suppose I have some operator $\mathcal{O}$ with eigenvalues $\lambda_n$ and the corresponding zeta function $$ \zeta_\mathcal{O}(s) = \sum_{n=1}^\infty \lambda_n^{-s} \,.$$ I want to compute the value of (the analytic continuation of) this function at zero. In the case of the Riemann zeta function, I've seen this done by establishing a functional relation between $\zeta(s)$ and $\zeta(1-s)$. However, I'm not sure how to generalise this to arbitrary spectral zeta functions. I suspect that analytically continuing this function throughout the complex plane is generally very hard (although I've never actually studied methods for explicitly computing analytic continuations; perhaps somebody could direct me to a resource), but I am only interested in the value at zero, and I thought perhaps this would be more feasible.

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