Show that $\displaystyle \Gamma(z)$ has no zeros.

I am not sure how to prove this. I would appreciate a few hints or suggestions. $\displaystyle \Gamma(z)$ is the gamma function. The gamma function is meromorphic. It is defined in the right half-plane by $\displaystyle \Gamma(z)= \int_0^{\infty} e^{-t}t^{z-1}dt$ for $\displaystyle \text{Re}(z)>0$. There is also another representation of $\displaystyle \Gamma(z)=\frac{\Gamma(z+m)}{(z+m-1) \cdots (z+1)z}$ where the right-hand side is defined and meromorphic for $\displaystyle \text{Re}(z)>-m$ with simple poles at $\displaystyle z=0, -1, \ldots, -m+1$. Thank you.