On discrete mean values of Dirichlet $L$-functions

Let χ be a nonprincipal Dirichlet character modulo a prime number p ≽ 3 and let $${\mathfrak{a}_{\cal X}}: = {1 \over 2}\left( {1{ - _{\cal X}}\left( { - 1} \right)} \right)$$ . Define the mean value $${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,: = {2 \over {p - 1}}\,\sum\limits_{\matrix{ {\psi \,\left( {\bmod \,p} \right)} \cr {\psi \left( { - 1} \right) = - 1} \cr } } {L\,\left( {1,\psi } \right)L\left( { - s,{\cal X}\overline \psi } \right)\,\,\,\,\left( {\sigma : = \,\,{\Re _s} > 0} \right).} $$ We give an identity for $${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\,$$ which, in particular, shows that $${{\cal M}_p}\left( { - s,{\cal X}} \right)\,\, = L\left( {1 - s{,_{\cal X}}} \right) + {\mathfrak{a}_{\cal X}}2{p^s}L\left( {1{,_{\cal X}}} \right)\zeta \left( { - s} \right) + o\left( 1 \right)\,\,\,\left( {p \to \infty } \right)$$ for fixed $$0 < \sigma < {1 \over 2}$$ and $$\left| {t: = \,{\mathfrak{J}_s}} \right| = o\left( {{p^{\left( {1 - 2\sigma } \right)/\left( {3 + 2\sigma } \right)}}} \right)$$ .