Let F be a non-Archimedean locally compact field of residual characteristic p with Weil group W F . Let σ be an irreducible smooth complex representation of W F , realized as the Langlands parameter of an irreducible cuspidal representation π of a general linear group over F . In an earlier paper, we showed that the ramification structure of σ is determined by the fine structure of the endo-class Θ of the simple character contained in π, in the sense of Bushnell-Kutzko. The connection is made via the Herbrand function Ψ Θ of Θ. In this paper, we concentrate on the fundamental Carayol case in which σ is totally wildly ramified with Swan exponent not divisible by p. We show that, for such σ, the associated Herbrand function satisfies a certain functional equation, and that this property essentially characterizes this class of representations. We calculate Ψ Θ explicitly, in terms of a classical Herbrand function arising naturally from the classification of simple characters. We describe exactly the class of functions arising as Herbrand functions Ψ Ξ , as Ξ varies over the set of totally wild endoclasses of Carayol type. In a separate argument, we derive a complete description of the restriction of σ to any ramification subgroup and hence a detailed interpretation of the Herbrand function. This gives concrete information concerning the Langlands correspondence.2000 Mathematics Subject Classification. 22E50, 11S37, 11S15.