  We investigate evolution of properties of an extended $s$-wave
superconductor, when the order parameter varies from an $s$-wave to a $g$-wave
continuously, by using a model order parameter $\Delta(\hat{k}) =
\Delta_{0}((1-x)+x \sin^{4}\theta \cos 4\phi)$. The evolution of the gap
amplitude, the density of states, and the specific heat are mainly focused on.
For $x<0.5$, due to the existence of a finite sized gap, the characteristic
behaviors more or less follow those of the $s$-wave. Sudden changes in the
characteristic behaviors come out for $x \ge 0.5$, due to appearances of nodes.
For $x=0.5$, point nodes in the order parameter on the Fermi surface appear,
while for $x>0.5$, line nodes appear. Although they are different kinds of
nodes which would usually induce different power-law dependencies in
superconducting properties, interestingly enough, they give rise to the same
characteristic behavior. The detailed structure of the point nodes for $x=0.5$
is investigated, and it is explained why they lead to the same dependence as
the line nodes.
