Functional equation for the zeta function

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The functional equation for Riemann zeta function is a result due to analytic continuation of Riemann zeta function:

\[\zeta(s)=2^s\pi^{s-1}\sin\left(\pi s\over2\right)\Gamma(1-s)\zeta(1-s)\]

There are multiple proofs for the functional equation for Riemann zeta function, and this page presents a lightweighted approach which merely relies on Fourier series that

\[\frac12-\{x\}=\sum_{n=1}^\infty{\sin(2\pi nx)\over\pi n}\]

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