We used a combination of polarized Raman spectroscopy and spin wave calculations to study magnetic excitations in the strong spin-orbit-coupled (SOC) bilayer perovskite antiferromagnet $Sr_3Ir_2O_7$. We observed two broad Raman features at ~ 800 $cm^{-1}$ and ~ 1400 $cm^{-1}$ arising from magnetic excitations. Unconventionally, the ~ 800 $cm^{-1}$ feature is fully symmetric ($A_{1g}$) with respect to the underlying tetragonal ($D_{4h}$) crystal lattice which, together with its broad line shape, definitively rules out the possibility of a single magnon excitation as its origin. In contrast, the ~ 1400 $cm^{-1}$ feature shows up in both the $A_{1g}$ and $B_{2g}$ channels. From spin wave and two-magnon scattering cross-section calculations of a tetragonal bilayer antiferromagnet, we identified the ~ 800 $cm^{-1}$ (~ 1400 $cm^{-1}$) feature as two-magnon excitations with pairs of magnons from the zone-center $Gamma$ point (zone-boundary van Hove singularity X point). We further found that this zone-center two-magnon scattering is unique to bilayer perovskite magnets which host an optical branch in addition to the acoustic branch, as compared to their single layer counterparts. This zone-center two-magnon mode is distinct in symmetry from the time-reversal symmetry broken spin wave gap and phase mode proposed to explain the ~ 92 meV (742 $cm^{-1}$) gap in RIXS magnetic excitation spectra of $Sr_3Ir_2O_7$.
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