Euler-Mascheroni Integrals

Define

$$I_n\equiv (-1)^n \int_0^\infty (\ln z)^n e^{-z}\,dz,$$ (1)

then

$$I_0 = \int_0^\infty e^{-z}\,dz = [-e^{-z}]^\infty_0 = (0+1)=1$$ (2)

$$I_1 = -\int_0^\infty (\ln z)e^{-z}\, dz = \gamma$$ (3)

$$I_2 = \gamma^2+{\textstyle{1\over 6}}\pi^2$$ (4)

$$I_3 = \gamma^3+{\textstyle{1\over 2}}\gamma\pi^2+2\zeta(3)$$ (5)

$$I_4 = \gamma^4+\gamma^2\pi^2-{\textstyle{3\over 20}}\pi^4+8\gamma\zeta(3),$$ (6)

where $\gamma$ is the Euler-Mascheroni Constant and $\zeta(3)$ is Apéry's Constant.