The Gamma duplication formula reads
$$ F(s) := \frac{\Gamma(s)\Gamma(s + 1/2)}{\Gamma(2s)} = \sqrt{\pi} 2^{1 - 2s} $$
This is an entire function of order 1.
Also, it does not vanish anywhere. Hence if we already knew that it was order $< 2$, by Hadamard's factorization theorem we could write
$$ F(s) = e^{As + B} $$
and then evaluating $F$ at $0$ and $1/2$ we get the duplication formula. Is there any way to easily see that $F$ is order $<2$? It's straightforward for $\text{Re}(s) > 1/2$ but for the other side I keep on coming to some statement like "$\sin (\pi s) \Gamma(s)$ is bounded below by $1/n!$ in neighborhoods $U_n \ni -n$ of width uniform in $n$", which I'm not sure how to deal with.