  In this paper, for any positive integer $\ell\geq2,$ we define
$\ell$-generalized Fibonacci zeta function. We then study its analytic
continuation to the whole complex plane $\mathbb{C}.$ 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 $\ell$-generalized
Fibonacci zeta function at negative integer arguments are rational.

Number Theory