On the integral representations of |Γ(z)|² and its Fourier transform

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We derive integral representations in terms of the Macdonald functions for the square modulus s | Γ(a + i s) |² of the Gamma function and its Fourier transform when a<0 and a= -1,-2,, generalizing known results in the case a>0. This representation is based on a renormalization argument using modified Bessel functions of the second kind, and it applies to the representation of the solutions of the Fokker-Planck equation.