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A simple L\'evy-$\alpha$ stable (SL) model is used to describe the data onelastic $pp$ and $p\bar p$ scattering at low-$|t|$ from SPS energies up to LHCenergies. The SL model is demonstrated to describe the data with a strongnon-exponential feature in a statistically acceptable manner. The energydependence of the parameters of the model is determined and analyzed. TheL\'evy $\alpha$ parameter of the model has an energy-independent value of 1.959$\pm$ 0.002 following from the strong non-exponential behavior of the data. Westrengthen the conclusion that the discrepancy between TOTEM and ATLAS elastic$pp$ differential cross section measurements shows up only in the normalizationand not in the shape of the distribution of the data as a function of $t$. Wefind that the slope parameter has different values for $pp$ and $p\bar p$elastic scattering at LHC energies. This may be the effect of the odderonexchange or the jump in the energy dependence of the slope parameter in theenergy interval 3 GeV $\lesssim \sqrt s \lesssim$ 4 GeV.

Simple L\'evy-$\alpha$ stable model analysis of elastic $pp$ and $p\bar p$ low-$|t|$ data from SPS to LHC energies