The following ratio:


$$\frac{\Gamma(2/5)^3}{\pi\Gamma(1/5)}$$


has kept appearing in my research, and the only thing I know about its value is that it is $\cong 0.7567213$, whence the following two questions:


Is the value of this ratio an algebraic number?


What is the exact value of this ratio?



Algebraicity of a ratio of values of the Gamma function
This number is expected to be transcendental. [This answer](https://mathoverflow.net/a/344203/6506) gives a conceptual framework for studying the algebraicity of such $\Gamma$ ratios, and in fact a completely explicit criterion (which is only conjectural, when it comes to establishing transcendence).


Your number is equal to
\begin{equation\*}
\frac{\Gamma(2/5)^3}{\Gamma(1/5)^2 \Gamma(4/5)}
\end{equation\*}
up to multiplication by an algebraic number. This ratio is described by the vector of exponents $(-2,3,0,-1)$ and the criterion there is not met for $u=2$. In fancy terms, the Bernoulli distribution $B\_1$ does not vanish on this vector.


