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Let $\chi$ be a nonprincipal Dirichlet character modulo a prime number $p⩾3$ and let ${𝔞}_{\chi }:=\frac{1}{2}(1-\chi (-1))$. Define the mean value $${ℳ}_p(-s,\chi ):=\frac{2}{p-1}\sum \psi (modp)\psi (-1)=-1L(1,\psi )L(-s,\chi \overline{\psi })(\sigma :=\Re s>0).$$ We give an identity for ${ℳ}_p(-s,\chi )$ which, in particular, shows that $${ℳ}_p(-s,\chi )=L(1-s,\chi )+{𝔞}_{\chi }2p^{s}L(1,\chi )\zeta (-s)+o\left(1\right)(p\to \infty )$$ for fixed $0<\sigma <\frac{1}{2}$ and $|t:=\Im s|=o\left(p^{(1-2\sigma )/(3+2\sigma )}\right)$.