Transcendency of the determinant of the Riemann operator: on higher $K$-groups

In previous papers we investigated basic properties of the determinant $G_K(s)$ of the Riemann operator: $\mathcal{R}$ acting on ${⨁}_{n>1}{K}_n(A{)}_{\mathbb{C}}$, where $A$ is the integer ring of an algebraic number field $K$. The function $G_K(s)$ is defined as the regularized determinant $$G_K(s)=\mathrm{d}\mathrm{e}\mathrm{t}((sI-\mathcal{R})\mathrm{\mid }\underset{n>1}{⨁}{K}_n(A{)}_{\mathbb{C}})$$ with $\mathcal{R}\mathrm{\mid }{K}_n(A{)}_{\mathbb{C}}=\frac{1-n}{2}$. We showed that $G_K(s{)}^{-1}$ is essentially the so called gamma factors of Dedekind zeta function of $K$. In this paper we study the transcendency of $G_K(s)$ for some rational numbers $s$. The result depends on types of $K$. For example, we show that $G_K(\frac{1}{3})$ is a transcendental number if $K$ is a totally imaginary and $G_K(\frac{1}{2})$ is a transcendental number otherwise.