This paper investigates superconvergence properties of the local discontinuous Galerkin methods with generalized alternating fluxes for one-dimensional linear convection-diffusion equations.By the technique of constructing some special correction functions,we prove the (2k + 1)-th-order superconvergence for the cell averages,and the numerical traces in the discrete L2 norm.In addition,superconvergence of orders k + 2 and k + 1 is obtained for the error and its derivative at generalized Radau points.All the theoretical findings are confirmed by numerical experiments.