The module $D{f}^s$ for locally quasi-homogeneous free divisors

We find explicit free resolutions for the $\text{scr}D$-modules $\text{scr}D{f}^s$ and $\text{scr}D[s]f^s\mathrm{/}\text{scr}D[s]f^{s+1}$, where $f$ is a reduced equation of a locally quasi-homogeneous free divisor. These results are based on the fact that every locally quasi-homogeneous free divisor is Koszul free, which is also proved in this paper