Euler's formula for the Gamma function is

$$\Gamma(z)=\frac{1}{z} \prod_{n=1}^{\infty}\left(1+\frac{1}{n}\right)^{z}\left(1+\frac{z}{n}\right)^{-1}$$

Use Euler's formula to show

$$\frac{\Gamma(2 z)}{2^{2 z} \Gamma(z) \Gamma\left(z+\frac{1}{2}\right)}=C$$

where $C$ is a constant.

Evaluate $C$.

[Hint: You may use $\Gamma(z) \Gamma(1-z)=\pi / \sin (\pi z) .]$