(Please pardon the use of nonstandard terminology, as I know not the accepted names for my entities of interest.)
Some personal research I have been doing led me to consider series of the form $$\sum_{k=1}^\infty\frac1{(k^2+bk+c)^s}$$
for complex $s$ and real $b,c$. I have been provisionally calling them "quadratic Dirichlet series", in the sense that a quadratic polynomial appears in the denominator of the $k$-th term.
I want to ask if these have ever been previously studied in the literature, and if there are more established names for them. I am particularly interested in references regarding their (symbolic or numeric) evaluation, but any related paper would be appreciated.
An obvious easy case is when $b=c=0$, where it degenerates to the usual $\zeta$ function. Apart from that, I haven't been able to find any simpler closed forms for these sums.
The Epstein function seemed close, but the expressions in the denominator are multi-index quadratic forms as opposed to the simpler(?) case I have.
Bonus: if there are any studies on the "polylogarithmic" generalization $$\sum_{k=1}^\infty\frac{z^k}{(k^2+bk+c)^s}$$ I would like to see them too.