proving that $f(z)=1/\Gamma(z)$ is a entire function

Question

$\Gamma: \mathbb{C}\rightarrow \mathbb{C}$ is the gamma function and $g:\mathbb{C}\rightarrow \mathbb{C}$ such that $g(z)=\frac{1}{z}$ is the reciprocal.

We know that these functions aren't entire but $\Gamma \in \mathbb{H}(\mathbb{C}\backslash\{0,-1,...\})$ and $g\in \mathbb{H}(\mathbb{C}\backslash\{0\})$, so if i prove that $\Gamma $ has no zero is enough to say that $f=g\times \Gamma: \mathbb{C}\rightarrow \mathbb{C}$ is entire?