<p>Ultracold atomic gases provide a controllable system to investigate the universal scaling laws of three-body physics, that is how the three-body recombination depends on the two-body scattering length. Although these scaling laws have been carefully examined in the bosonic systems, their manifestation in fermionic systems remain elusive. In this study, we investigate the three-body atomic loss rate <i>L</i>₃ in a two-component ⁶Li Fermi gas with a negative scattering length <i>a</i> &lt; 0. Our experimental results indicate that the scaling law exists near the narrow <i>s</i>-wave Feshbach resonance, characterized by <i>L</i>₃ ∝ ∣<i>a</i>∣^2.60. This scaling is observed to be valid only within a certain range of a, breaking down at both extremes. On one end, in the unitary regime where ∣<i>a</i>∣ approaches infinity, the collisional rate becomes independent of <i>a</i>. On the other end, as <i>a</i> approaches zero, a change in the length scale occurs, resulting in <i>L</i>₃ ∝ <i>r</i><Stack> _vdw ^2.30 </Stack>, where the van der Waals length <i>r</i>_vdw becomes the dominate scaling length. This study resolves the puzzle of the scaling law on the Bardeen-Cooper-Schrieffer (BCS) side of a two-component Fermi gas, shedding light on the universal physics in few-body systems.</p>