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

$\mathcal H_m$ of degree $n$, where $H$ runs through semi-integral positive definite Hermitian

matrices of degree $n$, i.e. $H\in \Lambda_m({\mathcal O})$ over the integers ${\mathcal O}$

of an imaginary quadratic field $K$, where $q^H=\exp(2\pi i {\rm Tr}(HZ))$.

Analytic $p$-adic continuation of their zeta functions is constructed via $p$-adic measures, bounded

or growing. Previously this problem was solved for the Siegel modular forms.

Main result is stated in terms of the Hodge polygon $P_{H}(t):

[0,d]\to {\mathbb R}$ and the Newton polygon $P_N(t)=P_{N,p}(t):

[0,d]\to {\mathbb 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.

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