The fractional Navier-Stokes equations on a periodic domain $[0,L]3$ differ from their conventional counterpart by the replacement of the $-\nu \Delta u$ Laplacian term by $\nu sAsu$, where $A=-\Delta$ is the Stokes operator and $\nu s=\nu L2(s-1)$ is the viscosity parameter. Four critical values of the exponent $s\ge 0$ have been identified where functional properties of solutions of the fractional Navier-Stokes equations change. These values are,: $s=13$,; $s=34$,; $s=56$ and $s=54$. In particular: i) for $s>13$ we prove an analogue of one of the Prodi-Serrin regularity criteria; ii) for $s\ge 34$ we find an equation of local energy balance and; iii) for $s>56$ we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi-Serrin criterion for $s>13$ suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range .