  We investigate the connection between the local electron correlation and the
momentum dependence of the spin susceptibility and the superconducting gap
functions in Sr$_2$RuO$_4$, using density-functional theory combined with
dynamical mean-field theory. Adopting frequency-dependent twoparticle vertex
moves the zero energy spin susceptibility peaks towards the Brillouin zone
center, compared with random-phase approximation which basically retains the
peak positions closer to the Brillouin zone boundary as determined by the
Fermi-surface nesting. We find that dxy orbital plays a central role here via
its enhanced correlation strength. Solving the linearized Eliashberg equation
from this spin susceptibility, prime candidates of the superconducting gap
symmetry are a s-wave, along with a nearly degenerate d-wave solution, all in
spin singlet. Furthermore, another set of degenerate spin singlet gap functions
emerges, odd with respect to k-point as well as orbital exchanges. We show that
the stability of these gap functions are strongly dependent on the peak
position of the spin susceptibility in the Brillouin zone.
