= Kronecker limit formula
{c}
{title2=$\mathcal E(z,s)=\dfrac1{2(s-1)}+\dfrac{\gamma-\log(4\pi)}2-\log(\sqrt y|\eta(z)|^2)+O(s-1)$}
This normalization of the first Kronecker limit formula follows from the <Fourier expansion> of the <completed nonholomorphic Eisenstein series>. At $s=1$, $K_{1/2}(t)=\sqrt{\pi/(2t)}e^{-t}$, so the nonconstant terms sum to $-2\log|\prod_{m\geq1}(1-q^m)|$. The constant terms supply $\pi y/6-(\log y)/2+(\gamma-\log(4\pi))/2$. Combining them gives the <Dedekind eta function> expression, with $\gamma$ the <Euler--Mascheroni constant>.
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