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Based on Guillemin's work on gauged Lagrangian distributions, we willintroduce the notion of a poly-$\log$-homogeneous distribution as an approachto $\zeta$-functions for a class of Fourier Integral Operators which includescases of amplitudes with asymptotic expansion $\sum_{k\in\mathbb{N}}a_{m_k}$where each $a_{m_k}$ is $\log$-homogeneous with degree of homogeneity $m_k$ butviolating $\Re(m_k)\to-\infty$. We will calculate the Laurent expansion for the$\zeta$-function and give formulae for the coefficients in terms of the phasefunction and amplitude as well as investigate generalizations to theKontsevich-Vishik quasi-trace. Using stationary phase approximation, seriesrepresentations for the Laurent coefficients and values of $\zeta$-functionswill be stated explicitly. Additionally, we will introduce an approximationmethod (mollification) for $\zeta$-functions of Fourier Integral Operatorswhose symbols have singularities at zero by $\zeta$-functions of FourierIntegral Operators with regular symbols.

A generalized Kontsevich-Vishik trace for Fourier Integral Operators and the Laurent expansion of $\zeta$-functions