A coefficient of an asymptotic expansion of logarithms of determinants for classical elliptic pseudodifferential operators with parameters

For classical elliptic pseudodifferential operators $A(\lambda )$ of order $m>0$ with parameter $\lambda$ of weight $\chi >0$, it is known that $\log \det_\theta A(\lambda )$ admits an asymptotic expansion as $\lambda \to+\infty$. In this paper we show, with some assumptions, that the coefficient of $\lambda ^{-1/\chi }$ can be expressed by the value of a zeta function at 0 for some elliptic $\psi \operatorname{DO}$ on $M\times S^1$ multiplied by $\frac {m}{2}$.