<s>[INST] Product of Gamma functions I What is the value of the product of Gamma functions
\begin{align}
\prod_{k=1}^{8} \Gamma\left( \frac{k}{8} \right)
\end{align}
and can it be shown that the product
\begin{align}
\prod_{k=1}^{16} \Gamma\left( \frac{k}{8} \right) = \frac{ 3 \Gamma(11) \ \pi}{2^{19}} \zeta(2) \zeta(4) \approx \frac{\pi^{7}}{26}
\end{align}
is a valid result?
 [/INST] For the first question, you can simply calculate:
$$\prod_{k=1}^{8}\Gamma\left(\frac{k}{8}\right) = \prod_{k=1}^{8}\frac{8}{k}\cdot\frac{k}{8}! = \frac{8^8}{8!}\cdot\prod_{k=1}^{8}\frac{k}{8}!$$
Or alternatively (even simpler):
$$\prod_{k=1}^{8}\Gamma\left(\frac{k}{8}\right) = \prod_{k=1}^{8}\frac{k-8}{8}!$$


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