The Euler–Poincaré characteristic is a generalization of the cardinality (or counting) and its higher homological extension for singular varieties as a natural transformation (what could be put in as its “categorification”) is MacPherson’s Chern class transformation. This transformation furthermore has two main developments: a bivariant-theoretic analogue and a generating series of it, i.e. a zeta function. The motivic Hirzebruch class is a unified theory of the three well-known characteristic classes of singular varieties, i.e. the above MacPherson’s Chern class transformation, Baum–Fulton–MacPherson’s Riemann–Roch and Cappell–Shaneson’s -class transformation, which extends Goresky–MacPherson’s -class. In this paper we discuss a bivariant-theoretic analogue and a zeta function of the motivic Hirzebruch class.