Determinants of Riemann operators on Quillen's higher $K$-groups: periodicity

In a previous paper [KT] we introduced determinant of the Riemann operator on Quillen’s higher $K$-groups of the integer ring of an algebraic number field $K$. We showed that the determinant expresses essentially the inverse of the so called gamma factor of Dedekind zeta function of $K$. Here we study the periodicity of determinant. This comes from the famous “periodicity” of higher $K$ groups. This periodicity is analogous to Euler’s periodicity of gamma function $\mathrm{\Gamma }(x+1)=x\mathrm{\Gamma }(x)$. We investigate the “reflection formula” corresponding to Euler’s reflection formula $\mathrm{\Gamma }(x)\mathrm{\Gamma }(1-x)=\frac{\pi }{\sin (\pi x)}$ also.