$L^p$ spectral multipliers on the free group $N_{3,2}$

Let $L$ be the homogeneous sublaplacian on the 6-dimensional free 2-step nilpotent group $N_{3,2}$ on 3 generators. We prove a theorem of Mihlin-H"ormander type for the functional calculus of $L$, where the order of differentiability $s>6\mathrm{/}2$ is required on the multiplier.