The stringy Euler number and stringy E-function are interesting invariants of log terminal singularities introduced by Batyrev. He used them to formulate a topological mirror symmetry test for pairs of certain Calabi-Yau varieties and to show a version of the McKay correspondence. It is a natural question whether one can extend these invariants beyond the log terminal case. Assuming the minimal model program, we introduce very general stringy invariants, associated to "almost all" singularities, more precisely, to all singularities that are not strictly log canonical. They specialize to the invariants of Batyrev when the singularity is log terminal. For example, the simplest form of our stringy zeta function is, in general, a rational function in one variable, but it is just a constant (Batyrev's stringy Euler number) in the log terminal case.
Willem Veys. "Stringy zeta functions for ℚ-Gorenstein varieties." Duke Math. J. 120 (3) 469 - 514, 1 December 2003. https://doi.org/10.1215/S0012-7094-03-12031-1