For $Π\subset \mathbb{R}^2$ a connected, open, bounded set whose boundary is a finite union of disjoint polygons whose vertices have integer coordinates, the logarithm of the discrete Laplacian on $LΠ\cap \mathbb{Z}^2$ with Dirichlet boundary conditions has an asymptotic expansion for large $L$ involving the zeta-regularized determinant of the associated continuum Laplacian. When $Π$ is not simply connected, this result extends to Laplacians acting on two-valued functions with a specified monodromy class.