What is the analytic continuation of $$f(x)=\sum_{n=1}^\infty\frac{2K_1(2\sqrt{n^x})}{\sqrt{n^x}}$$
where $K_1$ is a modified Bessel function of the second kind.
This converges for real $x>0.$
I noticed it looks somewhat similar to an analytic continuation to the entire complex plane besides simple poles of a meromorphic function, of the spectral zeta function for explicitly known spectra, found in certain applications, however it's slightly different:
$$h(x)=\sum_{n=1}^\infty n^xK_{-x}(2n).$$
And is probably unrelated to $f(x)$.