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Analytic continuation of the Eisenstein series defined over Hecke and Fricke subgroups

It is well known that the (real analytic) Eisenstein series is defined, in the slash notation, as follows $$E_{s}(\tau) = \sum\limits_{\gamma\in\Gamma_{\infty}\backslash\text{SL}(2,\mathbb{Z})}\left.y^...