The authors investigate the Selberg type zeta function of two variables, defined by \[ Z_\Gamma(s,t)=\prod_{p \in \text{Prim}(\Gamma)}\;\prod_{n=0}^\infty\big(1-N(p)^{-s-n}\big)^{\binom{t+n-1}{n}}, \quad \operatorname{Re} s >1, \quad t \in \mathbb C \] which interpolates the higher Selberg zeta-function of \(\operatorname{rank}r\)
\[ Z_\Gamma(s)^{(r)}=\prod_{p \in \text{Prim}(\Gamma)}\;\prod_{n_1,\dots,n_{r-1}=0}^\infty \big(1-N(p)^{-s-n_1-\dots-n_{r-1}}\big). \] Here \(\text{Prim}(\Gamma)\) is the set of the primitive hyperbolic conjugacy classes of discrete discrete co-compact torsion free subgroup \(\Gamma\) (on upper half-plane) of \(\text{SL}_2(\mathbb R)\), and \(N(p)\) is a square of the larger eigenvalues of \(p \in \text{Prim}(\Gamma)\).
In the paper, the analytic continuation of the Selberg type zeta-function of two variables in the cases as \(t=n \in\mathbb Z\) and \(t \not \in\mathbb Z\), and the determinant expression of the function \(Z_\Gamma(s,t)\) via the Laplacian on a Riemann surface are obtained. Also, the functional equation for \(t \in \mathbb Z\) is proved.