The mean value of \(|L(k,\chi)|^2\) at positive rational integers \(k \geq 1\) (Q2773258)

Let \(\chi\) denote a Dirichlet character modulo \(f>2\), so that there are \(\varphi(f)\) such characters. The author establishes the following mean-square formula for the \(L\)-functions \(L(s,\chi)\) at the positive integer \(s=k\), over characters satisfying \(\chi(-1)= (-1)^k\): NEWLINE\[NEWLINE{2\over \varphi(f)} \sum_{\chi(-1) =(-1)^k} \bigl|L(k,\chi) \bigr|^2 ={\pi^{2k} \over 2\bigl((k-1)! \bigr)^2} \sum^{2k}_{l=1} r_{k,l}\phi_l (f)f^{l-2k},NEWLINE\]NEWLINE where \(\phi_l(f)= \prod_{p |f}(1-p)\), and the coefficients \(r_{k,l}\) are given by the formula NEWLINE\[NEWLINE \sum^{2k}_{l-0} r_{k,l}d^l= \sum^{d-1}_{l=1} \bigl(\text{cot}^{(k-1)}(\pi l/d) \bigr)^2,NEWLINE\]NEWLINE with \(d>1\) being an integer. Thus, in the particular case \(k=4\), the `even' mean-value is given by NEWLINE\[NEWLINE{\pi^8\phi_8 (f)\over 9450}+ {\pi^8\phi_4 (f) \over 2025f^4}+ {\pi^8\phi_2(f) \over 567f^6}.NEWLINE\]