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The infinite grid is the Cayley graph of $\mathbb{Z} \times \mathbb{Z}$ withthe usual generators. In this paper, the Ihara zeta function for the infinitegrid is computed using elliptic integrals and theta functions. The zetafunction of the grid extends to an analytic, multivalued function whichsatisfies a functional equation. The set of singularities in its domain isfinite.  The grid zeta function is the first computed example which is non-elementary,and which takes infinitely many values at each point of its domain. It is alsothe limiting value of the normalized sequence of Ihara zeta functions forsquare grid graphs and torus graphs.

The Ihara zeta function of the infinite grid