We establish how the spectral decomposition for a Riemann surface determines the allocation of the bounded cohomology over the representations of SL₂(R). Then we explore the connections of the Dilogarithm with the continuous bounded cohomology of SL₂(R) and SL₂(C). In particular, it appears that Rogers' Dilogarithm is uniquely determined even measurably by the Spence–Abel functional equation.