Geodesic
Analogues of the Gauss--Vinogradov formula on the critical line
Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IV, Tome 109 (1981), pp. 41-82

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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. 
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     author = {A. I. Vinogradov and L. A. Takhtadzhyan},
     title = {Analogues of the {Gauss--Vinogradov} formula on the critical line},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {41--82},
     publisher = {mathdoc},
     volume = {109},
     year = {1981},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_1981_109_a2/}
}
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A. I. Vinogradov; L. A. Takhtadzhyan. Analogues of the Gauss--Vinogradov formula on the critical line. Zapiski Nauchnykh Seminarov POMI, Differential geometry, Lie groups and mechanics. Part IV, Tome 109 (1981), pp. 41-82. http://geodesic.mathdoc.fr/item/ZNSL_1981_109_a2/