Is the $\displaystyle \Gamma$ function uniquely determined (as a meromorphic function on $\displaystyle \mathbb{C}$) by the functional equation $\displaystyle \Gamma(s + 1) = s\Gamma(s)$ and a single value, say $\displaystyle \Gamma(1) = 1$?

And/or is there any other characterization of the Gamma function that uniquely determines it on $\displaystyle \mathbb{C}$?