The Euler product formula for the Gamma function is

$$\Gamma(z)=\lim _{n \rightarrow \infty} \frac{n ! n^{z}}{z(z+1) \ldots(z+n)}$$

Use this to show that

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

where $c$ is a constant, independent of $z$. Find the value of $c$.