= Completed nonholomorphic Eisenstein series
{title2=$\mathcal E(z,s)=\tfrac12\pi^{-s}\Gamma(s)\sum'_{m,n}y^s|mz+n|^{-2s}$}
This <modular-invariant function>, initially defined for $\Re s>1$, has <Fourier expansion>
$$
\mathcal E(z,s)=\Xi(2s)y^s+\Xi(2s-1)y^{1-s}+2\sqrt y\sum_{n\ne0}|n|^{s-1/2}\sigma_{1-2s}(|n|)K_{s-1/2}(2\pi|n|y)e^{2\pi inx},
$$
where $\Xi(u)=\pi^{-u/2}\Gamma(u/2)\zeta(u)$ is the <completed Riemann zeta function>, without the entire-xi polynomial factor. <Poisson summation> and the <integral representation of the modified Bessel function of the second kind> prove the expansion. It gives <meromorphic continuation>, $\mathcal E(z,s)=\mathcal E(z,1-s)$, and simple <poles> at zero and one with residues $-1/2,1/2$. Apparent poles at $s=1/2$ cancel between the two constant terms.
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