Wolfram-Alpha closed form of product of Gamma function

Question

Wolfram-Alpha recognizes this closed form \begin{align} \prod_{k=1}^{n-1}\sin(\pi k/n) &= 2^{1-n}\,n \end{align} just fine

but fails on this one,

despite that this expression also has a known closed form \begin{align} \prod_{k=1}^{n-1}\Gamma(k/n) &= \sqrt{\frac{ (2\,\pi)^{n-1}}{n}} . \end{align}

Question: Is there a way to make Wolfram-Alpha to recognize it?

Answer 1

Thanks to the comment by @Moo, I checked that it is indeed the Mathematica's fault and asked the follow-up question and get a quick response on how to mitigate this. For some reason, a cosmetic change of the expression \begin{align} \Gamma \left(\frac{k}{n}\right) &=\frac{\Gamma \left(\frac{k}{n}+1\right)}{\frac{k}{n}} \end{align}

makes the Mathematica and hence, Wolfram happy: