  We associate with an infinite cyclic cover of a punctured neighborhood of a
simple normal crossing divisor on a complex quasi-projective manifold (assuming
certain finiteness conditions are satisfied) a rational function in $K_0({\rm
Var}^{\hat \mu}_{\mathbb{C}})[\mathbb{L}^{-1}]$, which we call {\it motivic
infinite cyclic zeta function}, and show its birational invariance. Our
construction is a natural extension of the notion of {\it motivic infinite
cyclic covers} introduced by the authors, and as such, it generalizes the
Denef-Loeser motivic Milnor zeta function of a complex hypersurface singularity
germ.
