  Let $\left\{E_n\right\}_{n=1}^{\infty}$ be the set of energy levels
corresponding to a Hamiltonian $H$. Denote by
$$\lambda_{0}=0~~\textrm{and}~~\lambda_{n}=E_{n}$$ for $n\in\mathbb N.$ In this
paper, we shall construct and investigate the $p$-adic counterparts of the
Hurwitz-type spectral zeta function \begin{equation}
\zeta^{H}(s,\lambda)=\sum_{n=0}^{\infty}\frac{1}{(\lambda_{n}+\lambda)^{s}}
\end{equation} and its alternating form \begin{equation}
\zeta_{E}^{H}(s,\lambda)=2\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(\lambda_{n}+\lambda)^{s}}
\end{equation} in a parallel way.
