Dirichlet L-function

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Let $\chi$ be a Dirichlet character. The Dirichlet $L$-function associated with $\chi$ is $$L(s,\chi)=\displaystyle\sum_n \dfrac{\chi(n)}{n^s} = \displaystyle\prod_{p \hspace{2pt} \mathrm{prime}} \dfrac{1}{1-\chi(p)p^{-s}}.$$

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How Euler discovered the zeta function