Explicit formulae for the mean value of products of values of Dirichlet $L$-functions at positive integers

Abstract

Let $m\ge 1$ be a rational integer. We give an explicit formula for the mean value $$\frac{2}{\varphi (f)}\sum _{\chi (-1)=(-1{)}^m}|L(m,\chi ){|}^2,$$ where $\chi$ ranges over the $\varphi (f)/2$ Dirichlet characters modulo $f>2$ with the same parity as $m$. We then adapt our proof to obtain explicit means values for products of the form $L(m_1,{\chi }_1)\cdots L(m_{n-1},{\chi }_{n-1})\overline{L(m_n,{\chi }_1\cdots {\chi }_{n-1})}$.