Some equations involving the Gamma function

Let $V\subseteq\CC^{2n}$ be an algebraic variety with no constant coordinates and with a dominant projection onto the first $n$ coordinates. We show that the intersection of $V$ with the graph of the $\Gamma$ function is Zariski dense in $V$. Our method gives an explicit description of the distribution of these intersection points, and can be adapted for some other functions.