On the motivic class of the stack of bundles

Let G be a split connected semisimple group over a field. We give a conjectural formula for the motivic class of the stack of G-bundles over a curve C, in terms of special values of the motivic zeta function of C. The formula is true if C=P1 or G=SLn. If k=C, upon applying the Poincaré or called the Serre characteristic by some authors the formula reduces to results of Teleman and Atiyah–Bott on the gauge group. If k=Fq, upon applying the counting measure, it reduces to the fact that the Tamagawa number of G over the function field of C is |π1(G)|.