How can I evaluate the following Mellin transform $$ \int_{0}^{\infty}x^{s - 1}\operatorname{K}_{0}\left(x\right)^{n}\,{\rm d}x\quad\mbox{for positive integers}\ n\ ? $$ For $n = 1,2$, it is expressible in terms of Gamma functions.
Does this hold for all powers, or is the answer expressed in terms of different functions? What about for specific cases $n = 3, 4, 5$?