Hurwitz zeta function has definition
$$\zeta(s,q) = \sum_{n=0}^{\infty} \frac{1}{(q+n)^s}$$
Consider generalization
$$\tilde{\zeta}(s,q, t) = \sum_{n=0}^{\infty} \frac{1}{(q+n^t)^s}$$
Is $\tilde{\zeta}(s,q, t)$ known, at least for some $t \ne 1$, in closed form or in terms of known special function?