In my work arose the following series: $$g(s) = \sum_{n = 0}^\infty \frac{\log(\zeta(n+2))}{n+2}s^{n}.$$ It has radius of convergence $2$ and converges for $|s| \leq 2$ except at $s = 2$. I'm wondering, does anyone know any particular explicit values of this function besides $g(0) = \frac{1}{2}\log(\zeta(2))$, e.g., what is $g(1)$, $g(-1)$, or $g(-2)$? Also, does it have meromorphic continuation, and what is its behavior at the singularity $s = 2$?
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2$\begingroup$ Can you use $\log\zeta(n+2)\approx 1/2^{n+2}$, subtract off that explicit contribution, then the same with the $1/3^{n+2}$ term, etc.? $\endgroup$– MyNinthAccountCommented Nov 19, 2019 at 1:40
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