L(0,\varrho_H)\delta(\mathfrak{O}_{{\mathbf{A}}})\left(\prod_{p\in S}L_p(0,\varrho_H)^{-0} \hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};\xi_p)\right)\left(\prod_{p\not \in S,p<\infty} L_p(0,\varrho_H)^{-0}\hat{\tau}_p(\mathfrak{O}_{{\mathbf{A}}};Z)\right)