The Mayer–Ruelle operator conjecture for complex multiplication lattices

Let $L_{CM}={\mathbb Z}+\tau_D{\mathbb Z}$ be a lattice with complex multiplication by the ring of integers in the imaginary quadratic field ${\mathbb Q}(\sqrt{-D})$. Let $\hat H_D(s)\in {\cal A}_{\theta}\otimes {\cal K}$ satisfy $\operatorname{Tr}(\hat H_D(s))=\sum_{i=1}^{\infty}\lambda_i s^i$, where the $\lambda_i$ are real numbers representing the spectrum of the self-adjoint operator $\hat H_D$. Let $\mathcal L_D(s)$ denote the corresponding Mayer–Ruelle operator. The Mayer–Ruelle operator conjecture. One has

$$e^{-\hat H_D(s)}\equiv \mathcal L_D(s)$$

whenever $\Re(s)>\frac{1}{2}$. This conjecture proposes a link between the spectral data of a self-adjoint operator associated with a complex multiplication lattice and the Mayer–Ruelle operator; the source provides no resolution of the claim.