I can prove that
$$ \zeta(2) = \frac{\sqrt6}{2} \prod_{n\geq1} \frac{3n3n}{(3n+1)(3n-1)}\frac{4n4n}{(4n+1)(4n-1)}, $$
where $\zeta(k)$ denotes the Riemann zeta function. I also have other similar formulas for other $\zeta(2k)$. Is this worth anything to anyone, or are such formulas already well-known?
The product on the RHS is $$\prod_{n=1}^\infty\frac1{1-1/(3n)^2}\frac1{1-1/(4n))^2}.$$ Using $$\prod_{n=1}^\infty\left(1-\frac{z^2}{n^2}\right)=\frac{\sin\pi z}{\pi z}$$ gives this as equal to $$\frac{\pi^2}{3\sqrt6}.$$ Restoring the factor $\sqrt6/2$ gives $\pi^2/6=\zeta(2)$.