Ehrhart theory measures a polytope P discretely by counting the lattice points inside its dilates P, 2P, 3P, . . .. We compute the Ehrhart quasipolynomials of the standard Coxeter permutahedra for the classical Coxeter groups, expressing them in terms of the Lambert W function. A central tool is a description of the Ehrhart theory of a rational translate of an integer zonotope.