Can anyone compute $$\prod_{i=1}^{\infty}{1+(\frac{k}{i})^3}$$ for integer k? Can it be done in closed form, using only elementary functions, without the use of the Gamma function? For k=1, the closed expressions are known to include $\cosh(.)$ and $\pi$.
I hope to use this one day to find a closed expression of $\zeta(3)$. If we could compute $$\prod_{i=1}^{\infty}{1+(\frac{z}{i})^3}$$ for complex z, it follows that $\zeta(3)$ is the coefficient for $z^3$ in its expansion.