Let D = (V,E) be a primitive digraph. The exponent of D, denoted by γ(D), is the least integer k such that for any u, v ∈ V there is a directed walk of length k from u to v. The local exponent of D at a vertex u ∈ V, denoted by expD (u), is the least integer k such that there is a directed walk of length k from u to v for each v ∈ V. Let V = {1,2,... ,n}. Following [1], the vertices of V are ordered so that exPD (1) ≤expD (2)≤…≤expD (n) ＝γ(D). Let En(i) :＝ {expD (i) | D ∈ PDn}, where PDn is the set of all primitive digraphs of order n. It is known that En(n) = {γ(D) | D ∈ PDn} has been completely settled by [7]. In 1998, En(1)was characterized by [5]. In this paper, the authors describe En(2) for all n≥2.