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

We find explicit free resolutions for the $scr D$-modules ${scr D} f^s$ and ${scr D}[s] f^s/{scr D}[s] f^{s+1}$, 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