  We construct a new equation of state (EOS) for numerical simulations of
core-collapse supernovae and neutron-star mergers based on an extended
relativistic mean-field model with a small symmetry energy slope $L$, which is
compatible with both experimental nuclear data and recent observations of
neutron stars. The new EOS table (EOS4) based on the extended TM1 (TM1e) model
with $L=40$ MeV is designed in the same tabular form and compared with the
commonly used Shen EOS (EOS2) based on the original TM1 model with $L=110.8$
MeV. This is convenient and useful for performing numerical simulations and
examining the influences of symmetry energy and its density dependence on
astrophysical phenomena. In comparison with the TM1 model used in EOS2, the
TM1e model provides a similar maximum neutron-star mass but smaller radius and
tidal deformability for a $1.4 M_\odot$ neutron star, which is more consistent
with current constraints. By comparing the phase diagram and thermodynamic
quantities between EOS4 and EOS2, it is found that the TM1e model predicts
relatively larger region of nonuniform matter and softer EOS for neutron-rich
matter. Significant differences between EOS4 and EOS2 are observed in the case
with low proton fraction, while the properties of symmetric matter remain
unchanged.
