  A simple L\'evy-$\alpha$ stable (SL) model is used to describe the data on
elastic $pp$ and $p\bar p$ scattering at low-$|t|$ from SPS energies up to LHC
energies. The SL model is demonstrated to describe the data with a strong
non-exponential feature in a statistically acceptable manner. The energy
dependence of the parameters of the model is determined and analyzed. The
L\'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. We
strengthen the conclusion that the discrepancy between TOTEM and ATLAS elastic
$pp$ differential cross section measurements arises only in the normalization
and not in the shape of the distribution of the data as a function of $t$. We
find 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 odderon
exchange or the jump in the energy dependence of the slope parameter in the
energy interval 3 GeV $\lesssim \sqrt s \lesssim$ 4 GeV.
