  In this paper, we study weighted low-lying zeros of spinor and standard
$L$-functions attached to degree 2 Siegel modular forms. We show the symmetry
type of weighted low-lying zeros of spinor $L$-functions is symplectic, for
test functions whose Fourier transform have support in $(-1,1)$, extending the
previous range $(-\frac{4}{15},\frac{4}{15})$ by E. Kowalski, A. Saha and J.
Tsimerman . We then show the symmetry type of weighted low-lying zeros of
standard $L$-functions is also symplectic. We further extend the range of
support by performing an average over weight. As an application, we discuss
non-vanishing of central values of those $L$-functions.
