I have seen Stirling's formula often stated as
$$\Gamma(s) = \left(\frac{2\pi}{s}\right)^{1/2}\left(\frac{s}{e}\right)^s(1 + O(|s|^{-1}))$$ with the implied constant absolute for $\Re s > 0$ (I think this can be made larger, though it isn't relevant here). However, how don't the $s^{-1/2}$ and $s^s$ that occur make it ambiguous since one has to specify a branch?