The spectral zeta function is defined as:
$$\zeta(s)=\sum_{n=1}^\infty \lambda_{n}^{-s}$$
Where $\lambda_n$ is the real spectrum of $L$ (an elliptic self adjoint operator).
I derived an analytic continuation of $\zeta(s)$:
$$ \Psi(s)=\sum_{n=1}^\infty 2\sqrt{n^s}K_1(2\sqrt{n^s}) $$
where $K_1$ is a Bessel function.
Now this form shows up in QFT when computing the one-loop effective action. Essentially you use the spectral zeta function and analytically continue it getting $\Psi(s).$
I derived the analytic continuation in just one line using my own approach, but I'm wondering what is the usual process to obtain $\Psi(s)$ from $\zeta(s)?$
See this for more: https://arxiv.org/pdf/1411.7069.pdf.