I am obviously overgeneralizing.........

The moduli of the complex factorials of real negative numbers, and imaginary numbers are equal to their respective real positive number factorials. ... The proposed concept has also been extended to Euler's gamma function for real negative numbers and imaginary numbers, and beta function.Nov 6, 2014

Factorials of real negative and imaginary numbers - A new perspective

https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4247832/

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complex analysis - On the modulus of $\Gamma(z)$ - Mathematics ...

https://math.stackexchange.com/questions/114600/on-the-modulus-of-gammaz

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Feb 28, 2012 - Added. For Re(z)=x>0 the Gamma integral converges ... to the function w(t)=e−ttx+iy−1=e−ttx−1tiy and observe that. |tiy|=|eiylogt|=1, ...