Analytic continuation of $\mathrm{\ell }$-generalized Fibonacci zeta function

In this paper, for any positive integer $\mathrm{\ell }\ge 2,$ we define $\mathrm{\ell }$-generalized Fibonacci zeta function. We then study its analytic continuation to the whole complex plane $\mathbb{C}\mathrm{.}$ Further, we compute a possible list of singularities and residues of the function at these simple poles. Moreover, we deduce that the special values of $\mathrm{\ell }$-generalized Fibonacci zeta function at negative integer arguments are rational.