The Gamma function satisfies:

$$\Gamma(z + 1) = z\Gamma(z)$$ $$\Gamma(z)\Gamma(1 - z) = \frac{\pi}{\sin(\pi z)}$$ $$\prod_{k = 0}^{n-1}\Gamma\left(a+\frac{k}{n}\right) = (2\pi)^{(n-1)/2}n^{-na + 1/2}\Gamma(na)$$

The Rohrlich conjecture states:

Any multiplicative dependence relation of the form $$\pi^{b/2}\prod_{a \in \mathbb{Q}}\Gamma(a)^{m_a} \in \bar{\mathbb{Q}}$$ is a consequence of the relations above.

Do we know special cases where this is valid? For instance, given rationals $a$ and $b$, do we know when $\dfrac{1}{\pi}\dfrac{\Gamma(a)\Gamma(b)}{\Gamma(a + b)}$ is algebraic? And for three rationals?