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In the Basel problem, the $sinc$ function is considered at the Wikipedia page. Let me try to make an alternative function definition: $$f(x) = \prod_{n=1}^\infty \left ( 1+ \frac{x^3}{n^3} \right ) = \...

In order to see what happens when taking the functional equation in this form: $$\xi(s) := \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s) $$ $$\xi(s) = \xi(1 - s)$$ $$\pi^{-s/2}\ \Gamma\left(\...