The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the Cauchy-Riemann operators $\det\overline{\partial}_j$ which act on the space of $j$-differentials in terms of Selberg zeta function?

The determinant of hyperbolic Maaß-Laplacian operator on arbitrary tensors and spinors can be written in terms of Selberg zeta function. Is there a corresponding formula for the determinant of the Cauchy-Riemann operators $\det\overline{\partial}_j$ which act on the space of $j$-differentials (i.e. arbitrary tensors and spinors) in terms of Selberg zeta function?