  Superconductivity in LaRu$_3$Si$_2$ with the honeycomb structure of Ru atoms
has been investigated. It is found that the normal state specific heat C/T
exhibits a deviation from the Debye model down to the lowest temperature. A
relation $C/T = \gamma_n+\beta T^2-ATlnT$ which concerns the electron
correlations can fit the data very well. The suppression to the
superconductivity by the magnetic field is not the mean-field like, which is
associated well with the observation of strong superconducting fluctuations.
The field dependence of the induced quasiparticle density of states measured by
the low temperature specific heat shows a non-linear feature, indicating the
significant contributions given by the delocalized quasiparticles.
