Geodesic
Approximate functional equation for the~product of two Dirichlet $L$-functions
Izvestiya. Mathematics , Tome 45 (1995) no. 2, pp. 255-280.

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An approximate functional is derived for $L(s,\chi_1)L(s,\chi_2)$, where $\chi_1$ and $\chi_2$ are primitive Dirichlet characters modulo $k_1$ and $k_2$, and also an approximate functional equation for an analogue of the Hardy–Selberg function. If $s=1/2+it$, $k_1k_2\leqslant |t|^{1/9 -5\varepsilon}$, then the remainder terms in these formulas are bounded by $O(|t|^{-\varepsilon})$ as $|t|\to\infty$ (where $\varepsilon$ is an arbitrarily small positive number). 
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S. A. Gritsenko. Approximate functional equation for the~product of two Dirichlet $L$-functions. Izvestiya. Mathematics , Tome 45 (1995) no. 2, pp. 255-280. http://geodesic.mathdoc.fr/item/IM2_1995_45_2_a1/

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