= Pole-subtracted theta integral for the completed zeta function
{title2=$\Lambda(s)=\pi^{-s/2}\Gamma(s/2)\zeta(s)$}
Let $J(s)=\int_1^\infty(\theta(t)-1)t^{s/2-1}\,dt$, with $\theta(t)=\sum_{n\in\mathbb Z}e^{-\pi n^2t}$. Exponential decay makes $J$ an <entire function>. The <Jacobi theta function> transformation and a substitution in the <Mellin transform> give
$$
\Lambda(s)=\frac1{s-1}-\frac1s+\frac12\bigl(J(s)+J(1-s)\bigr).
$$
This continues the completed <Riemann zeta function> meromorphically, with residues $1$ and $-1$ at $1$ and $0$, and makes the <Functional equation of the Riemann zeta function> immediate. Dividing by the <Gamma function> removes the apparent pole at zero from $\zeta(s)$.
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