An evaluation of the central value of the automorphic scattering determinant

Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\varphi (s)$ denote the automorphic scattering determinant. From the known functional equation $\varphi (s)\varphi (1-s)=1$ one concludes that $\varphi (1\mathrm{/}2{)}^2=1$. However, except for the relatively few instances when $\varphi (s)$ is explicitly computable, one does not know $\varphi (1\mathrm{/}2)$. In this article we address this problem and prove the following result. Let $N$ and $P$ denote the number of zeros and poles, respectively, of $\varphi (s)$ in $(1\mathrm{/}2,\mathrm{\infty })$, counted with multiplicities. Let $d(1)$ be the coefficient of the leading term from the Dirichlet series component of $\varphi (s)$. Then $\varphi (1\mathrm{/}2)=(-1{)}^{N+P}\cdot \mathrm{s}\mathrm{g}\mathrm{n}(d(1))$.