Why is the Gamma function shifted from the factorial by 1?

From Riemann's Zeta Function, by H. M. Edwards, available as a Dover paperback, footnote on page 8: "Unfortunately, Legendre subsequently introduced the notation $\Gamma(s)$ for $\Pi(s-1).$Legendre's reasons for considering $(n-1)!$ instead of $n!$ are obscure (perhaps he felt it was more natural to have the first pole at $s=0$ rather than at $s = -1$) but, whatever the reason, this notation prevailed in France and, by the end of the nineteenth century, in the rest of the world as well. Gauss's original notation appears to me to be much more natural and Riemann's use of it gives me a welcome opportunity to reintroduce it."

I would argue against the OP's opinion. The definition $\Gamma(z)$ becomes very natural if you write it as $\Gamma(z) = \int_0^\infty t^{z} e^{-t} d^\times t$, where $d^\times t=dt/t$ is the Haar measure on the multiplicative group of positive numbers. Moreover, $t^z$ is a character of this group, hence the definition is an instance of the Fourier transform on locally compact abelian groups, in this case called the Mellin transform. In fact, this is why this version works well for the Riemann zeta function and in fact for any automorphic $L$-function: $\pi^{-\frac{s}{2}}\Gamma(\frac{s}{2})\zeta(s)$ is invariant under $s\to 1-s$. Of course, one might say that $\zeta(s)$ is not normalized in the right way, but in terms of the Dirichlet coefficients of $\zeta(s)$, or more generally in terms of the Langlands parameters of an automorphic $L$-function, the current normalization is the right one (cf. Ramanujan conjecture)!

EDIT: I just realized this is an elaboration of a comment Emerton made earlier.

It was so that Legendre could do with the gamma function what the Catholic church did 170 years later: He put a simple pole at the origin.