Wikipedia mentions that the number $$a = \dfrac{\Gamma\left(\dfrac{1}{4}\right)}{\pi^{1/4}}$$ is transcendental. Since $\Gamma(1/2) = \sqrt{\pi}$ the above number $a$ seems to connected to a combination of $\Gamma(1/4)$ and $\Gamma(1/2)$. On squaring the connection is clear as we can see that $$a^{2} = \frac{\Gamma^{2}(1/4)}{\Gamma(1/2)} = B(1/4, 1/4)$$ where $B$ represents the beta function.
Are there any other known values of beta function which are transcendental? Any reference to a proof of the transcendence of number $a$ given above would also be helpful.