Let M be a finite volume, non-compact hyperbolic Riemann surface, possibly with elliptic fixed points, and let \(\chi \) denote a finite dimensional unitary representation of the fundamental group of M. Let \(\Delta \) denote the hyperbolic Laplacian which acts on smooth sections of the flat bundle over M associated with \(\chi \). From the spectral theory of \(\Delta \), there are three distinct sequences of numbers: the first coming from the eigenvalues of \(L^{2}\) eigenfunctions, the second coming from resonances associated with the continuous spectrum, and the third being the set of negative integers. Using these sequences of spectral data, we employ the super-zeta approach to regularization and introduce two super-zeta functions, \(\mathcal {Z}_-(s,z)\) and \(\mathcal {Z}_+(s,z)\) that encode the spectrum of \(\Delta \) in such a way that they can be used to define the regularized determinant of \(\Delta -z(1-z)I\). The resulting formula for the regularized determinant of \(\Delta -z(1-z)I\) in terms of the Selberg zeta function, see Theorem 5.3, encodes the symmetry \(z\leftrightarrow 1-z\).

令M为有限体积的非紧致双曲Riemann曲面，可能具有椭圆不动点，令\（\ chi \）表示M的基群的有限维表示。令\（\ Delta \）表示双曲型Laplacian，它作用于与\（\ chi \）相关联的M上的平坦束的光滑部分。根据\（\ Delta \）的谱理论，存在三个不同的数字序列：第一个序列来自\（L ^ {2} \）的特征值本征函数，第二个来自与连续光谱相关的共振，第三个是负整数的集合。利用这些光谱数据序列，我们采用了超级零点方法进行正则化，并引入了两个超级零点函数\（\ mathcal {Z} _-（s，z）\）和\（\ mathcal {Z} _ + （s，z）\）以这样的方式对\（\ Delta \）的频谱进行编码，即可以用来定义\（\ Delta -z（1-z）I \）的正则行列式。根据Selberg zeta函数，\（\ Delta -z（1-z）I \）的正则行列式的所得公式（见定理5.3）编码对称性\（z \ leftrightarrow 1-z \）。