  Ultracold atomic gases provide a controllable system to study the inelastic
processes for three-body systems, where the three-body recombination rate
depends on the scattering length scaling. Such scalings have been confirmed in
bosonic systems with various interaction strengths, but their existence with
fermionic atoms remains elusive. In this work, we report on an experimental
investigation of the scaling law for the three-body atomic loss rate $L_3$ in a
two-component $^6$Li Fermi gas with the scattering length $a<0$. The scaling
law is validated within a certain range of $a$ near the narrow $s$-wave
Feshbach resonance, where $L_3\propto T|a|^{2.60(5)}$, and $T$ is the gas
temperature. The scaling law is observed to have an upper and a lower bound in
terms of the scattering length. For the upper bound, when $a\rightarrow
\infty$, the power-law scaling is suppressed by the unitary behavior of the
resonance caused by the strong three-body collisions. For the lower bound,
$a\rightarrow 0$, the finite range effect modifies the scaling law by the
effective scattering length $L_e$. These results indicate that the three-body
recombination rate in a fermionic system could be characterized by the scaling
law associated with the generalized Efimov physics.
