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The multinomial Mittag-Leffler function plays a crucial role in the study of multi-term time-fractional evolution equations. In this work we establish basic properties of the Prabhakar type generalization of this function with the main emphasis on complete monotonicity. As particular examples, the relaxation functions for equations with multiple time-derivatives in the so-called "natural" and "modified" forms are studied in detail and useful estimates are derived. The obtained results extenddoi:10.1515/fca-2021-0005 fatcat:do2bwrdnivd35mczobnlr6z6mi