For a prime p and a positive integer n, the standard zeta function L_F (s) is considered, attached to an Hermitian modular form F =\sum_ H A(H)q^H on the Hermitian upper half plane H_n of degree n, where H runs through semi-integral positive definite Hermitian matrices of degree n, i.e. H \in \Lambda_n(O) over the integers O of an imaginary quadratic field K, where q^H = exp(2 iTr(HZ)). Analytic p- adic continuation of their zeta functions constructed by A.Bouganis in the ordinary case (in [Bou16] is presently extended to the admissible case via growing p-adic measures. Previously this problem was solved for the Siegel modular forms, [CourPa], [BS00]. Present main result is stated in terms of the Hodge polygon P_H(t) : [0; d] ! R and the Newton polygon P_N(t) = P_{N;p}(t) : [0; d] -> R of the zeta function L_F (s) of degree d = 4n. Main theorem gives a p-adic analytic interpolation of the L values in the form of certain integrals with respect to Mazur-type measures.