The binomial series gives a meromorphic continuation of any series of the form $\sum_{n=N}^\infty f(n)^{-s}$ for some polynomial $f(x) = \prod_{k=1}^d (x-a_k)$

$$f(n)^{-s} = n^{-ds} \prod_{k=1}^d (1-a_kn^{-1})^{-s} = n^{-ds} \prod_{k=1}^d \left(\sum_{m=0}^\infty {-s \choose m} (-a_k n)^{-sm}\right) \\= \sum_{l=0}^\infty h_l(s) n^{-s(d+l)}$$
$$\sum_{n=N}^\infty f(n)^{-s} = \sum_{l=0}^\infty h_l(s) \zeta_N(s(d+l)), \qquad \zeta_N(s) = \sum_{n=N}^\infty n^{-s}$$