Integer and Complex Values for the Gamma Function:

While reading the Wikipedia page on Particular values of the Gamma Function, it listed a formula: $$\Gamma\left(\dfrac z2\right)=\sqrt{\pi}\dfrac {(z-2)!!}{2^{(z-1)/2}}\tag{1}$$
Where $z\in\mathbb{Z}$ for positive half integers. $(1)$ can be used to compute $\Gamma\left(\frac 12\right)$ by setting $z=1$ to get $$\Gamma\left(\dfrac 12\right)=\sqrt{\pi}\dfrac {(1-2)!!}{2^{(1-1)/2}}=\sqrt\pi\tag{2}$$
Extending this, I'm wondering

Questions:

1. Can forumula $(1)$ be generalized to include complex numbers? $$\Gamma\left(a+bi\right)=\text{something}\tag{3}$$


2. If so, how would you prove such formula?

Running it for WolframAlpha, it says that the Gamma function of a complex number is defined and is possible. But I'm just not sure how to derive a formula for $(3)$.