Discontinuous Galerkin discretization of conservation laws in two dimensions by means of piecewise polynomials on unstructured meshes are considered. The applied concept for deriving an asymptotically exact error estimation rests on a local superconvergence at the outflow ends of the triangular elements. The authors give detailed proofs for the obtained error expansions and evaluate approximately the discretization error by the leading terms of its series representations. The related coefficients can be obtained via simple linear systems. Finally two numerical examples are given which show an excellent coincidence between theoretical estimates and practically occurring errors.