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In this article, we obtain transformation formulas analogous to the identityof Ramanujan, Hardy and Littlewood in the setting of primitive Maass cusp formover the congruence subgroup $\Gamma_0(N)$ and also provide an equivalentcriterion of the grand Riemann hypothesis for the $L$-function associated tothe primitive Maass cusp form over $\Gamma_0(N)$.

Equivalent criterion for the grand Riemann hypothesis associated to Maass cusp forms