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It can be extended to all $s$ by analytic continuation. For any integer $m$, the famous Gauss sum $G(m,\\chi)$ is defined as follows: $$G(m,\\chi)=\\sum_{a=1}^{q}\\chi(a)e\\left(\\frac{am}{q}\\right), $$ where $e(y)=e^{2\\pi iy}$. 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