The usual Riemann Zeta function is
$$ \zeta(s) = \sum_n \frac{1}{n^s} $$
Suppose we modify the denominator instead to
$$ \zeta_?(s) = \sum_n \frac{1}{(n\sqrt{1+Bn^2})^s} $$
We can do some elementary algebraic manipulations to get
$$ \zeta_?(s) = \sum_n \frac{1}{(n^2+Bn^4)^{s/2}} $$
And lastly, of course, we can make the substitution $B = C^4$ and scale the exponent by 2 to get
$$ \zeta_?(2s) = \sum_n \frac{1}{(n^2+(Cn)^4)^{s}} $$
Do any of these expressions match one of the many zeta functions that have been talked about in the literature?
For those curious how I got to this, it's because the Riemann zeta function happens to have an interesting interpretation in music theory as representing how well some equal division of the octave represents the harmonic series and simple rational intervals. But the piano timbre is slightly inharmonic, and follows the equation above, where $B$ is the inharmonicity coefficient, so this modified equation tells you how well some equal division of the octave represents the piano timbre (and thus intervals and chords derived from that timbre instead).
Of course this can be viewed as a generalized Dirichlet Series, which I think generalizes every one of these functions, but I'm curious if there's anything beyond that.