  It was observed recently that the multidimensional Mellin--Barnes integrals
(Gustafson's integrals) arise naturally in studies of the $SL(2,R)$ spin chain
models. We extend this analysis to the noncompact $SL(2,\mathbb{C})$ spin
magnets and obtain integrals which generalize Gustafson's integrals to the
complex case.
