Analogues of the Gauss–Vinogradov formula on the critical line

An asymptotic behavior of the sum $\sum_{p\equiv v(\operatorname{mod}4),\ p\le X}L(s,\chi_p)$ for $X\to\infty$ is studied in the critical strip, where $L(s,\chi_p)$ is the Dirichlet series with the quadratic character $\chi_p$ modulo $p$, where $p$ is a prime number; $v=1$ or $3$. With the help of large seive estimates a formula for this sum is obtained with two asymptotic terms on the critical line of the variable $s$. As a corollary the asymptotic expansion of this sum at the point $s=1/2$ is presented. The asymptotic formula for the sum $\sum_{|d|\le X}L(s,\chi_d)$, where $d$ runs over discriminants of quadratic fields, is also obtained.