Let MD s(n)={D|D is a primitive ministrong digraph with n vertrices, and the shortest cycle length of D is s}, and bs(n)=max {y(D)|D∈MDs(n)}, where y (D)is the primitivae exponent of D. Our main results are: (1) we give explicit exporessions for bs (n); (2) for s≠2, 6, we give a necessary and sufficient conditinon for a digraph D∈MDs (n) with y (D)=bs (n)