= Constant term of a nonholomorphic Eisenstein series
{title2=$A_0(y,s)$}
For the series over all nonzero integer pairs, the constant <Fourier coefficient> is $2\zeta(2s)y^s+2\sqrt\pi\,\Gamma(s-1/2)\zeta(2s-1)y^{1-s}/\Gamma(s)$. Integrating over one real period unfolds the translated intervals and then uses the <Gaussian integral>. Multiplication by $\pi^{-s}\Gamma(s)$ expresses it as $2\Lambda(2s)y^s+2\Lambda(2s-1)y^{1-s}$ in terms of the <completed Riemann zeta function>.
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