Cosmological model with Gong-Zong Parametrization in $f(R,L_m)$ gravity

We present the cosmic expansion scenarios in the $f(R, L_m)$ gravity studied by using the dark energy equation of state (EoS) parameters. We proceed with the specific form of $f(R, L_m)$ gravity termed as $f(R, L_m)=\frac{R}{2}+L_{m}^α$. We derive the expansion rate in terms of the red-shift for two different forms of EoS parameter. In first model, EoS parameter varies inversely with the redshift and in second model, it involves the exponential form with the redshift. By using the Bayesian methods based on the $χ^{2}$-minimization technique, the median values of model parameters are determined for the cosmic chronometer(CC) and Joint (CC+Pantheon) data sets. The behavior of fundamental cosmological parameters such as the deceleration parameter, energy density and pressure are thoroughly examined. Additionally, the nature of Statefinder diagnostics and the present age of universe exemplifies the compatibility with the late-time astronomical observations.