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From the known functional equation $\\phi(s)\\phi(1-s)=1$ one concludes that $\\phi(1/2)^{2} = 1$. However, except for the relatively few instances when $\\phi(s)$ is explicitly computable, one does not know $\\phi(1/2)$. 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 $\\phi(s)$ in $(1/2,\\infty)$, counted with multiplicities. 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Friedman, Lejla Smajlovic","submitted_at":"2016-07-27T11:54:26Z","abstract_excerpt":"Let $M$ be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let $\\phi(s)$ denote the automorphic scattering determinant. From the known functional equation $\\phi(s)\\phi(1-s)=1$ one concludes that $\\phi(1/2)^{2} = 1$. However, except for the relatively few instances when $\\phi(s)$ is explicitly computable, one does not know $\\phi(1/2)$. 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 $\\phi(s)$ in $(1/2,\\infty)$, counted with multiplicities. 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