How justify that $\Gamma\left(\frac{1}{6}\right)$ is a transcendental number

Question

Let $\Gamma(x)$ the Gamma function. See, if you need its definiton, for example in this Wikipedia.

While I was evaluating an integral with the help of Wolfram Alpha, I have known that $$\Gamma\left(\frac{1}{6}\right)$$ is a transcendental number.

Question. What's the reasoning to know this fact, that $\Gamma\left(\frac{1}{6}\right)$ is a trancendental number? Thanks in advance.

I presume that it is consequence of a theorem or computational method based on the definition of transcendental numbers, and that the Gamma function is a factorial. How justify that previous real number is trascendental?

Answer 1

In this answer the identity

$$\Gamma\left(\frac{1}{6}\right) = {\frac{2^{\frac{14}{9}}\cdot 3^{\frac{1}{3}}\cdot \pi^{\frac{5}{6}} }{\text{AGM}\left(1+\sqrt{3},\sqrt{8}\right)^{\frac{2}{3}}}}\tag{1}$$

is proved. The trascendence of $\Gamma\left(\frac{1}{6}\right)$ then follows from recalling classical results (Theorem $7$ here) about the trascendence of periods / complete elliptic integrals.