Explicit formulas for sums related to Dirichlet L-functions

Abstract

Let $p\geq3$ be a prime number and let $m, n$ and $l$ be integers with $\gcd(l,p)=1$. Let $\chi$ be a Dirichlet character modulo $p$ and $L(s,\chi)$ be the Dirichlet L-function corresponding to $\chi$. Explicit formulas for: $$\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=+1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi}) \text{ and }\dfrac{2}{p-1} \sum \limits\sb{\underset{\chi(-1)=-1}{\chi\hspace{-0.2cm} \mod p}} \chi(l) L(m,\chi)L(n,\overline{\chi})$$ are given in this paper by using the properties of character sums and Bernoulli polynomials.