What is $\Gamma(z) \Gamma(\bar {z})$?

Question

Is there an identity for $\Gamma(z) \Gamma(\bar {z})$?

Post-question edits:

Have you come across any, in addition to the ones listed here? (there, special cases, where the real part of $z$ is integer or half integer, are also listed.)

Abramowitz-Stegun lists the following under 6.1.45 $$\lim_{|y|\to \infty} (2\pi)^{-1/2} |\Gamma (z)| e^{\pi|y|/2} |y|^{1/2-x} =1$$

Answer 1

Since $\Gamma(\bar{z}) = \overline{\Gamma(z)}$ $$\Gamma(z)\Gamma(\bar{z})=\Gamma(z)\overline{\Gamma(z)} = |\Gamma(z)|^2$$

Answer 2

Let $z=x+iy$, $$\Gamma[z] \Gamma[\bar z|=|\Gamma(x+iy)|^2=\Gamma^2(x) \prod_{n=0}^{\infty} \left(1+\frac{y^2}{(x+n)^2} \right)^{-1}$$