  For a homogeneous polynomial of $n$ variables, we present a new method to
compute the roots of Bernstein-Sato polynomial supported at the origin,
assuming that general hyperplane sections of the associated projective
hypersurface have at most weighted homogeneous isolated singularities.
Calculating the dimensions of certain $E_r$-terms of the pole order spectral
sequence for a given integer $r\in[2,n]$, we can detect its degeneration at
$E_r$ for certain degrees. In the case of strongly free, locally positively
weighted homogeneous divisors on ${\mathbb P}^3$, we can prove its degeneration
almost at $E_2$ and completely at $E_3$ together with a symmetry of a modified
pole-order spectrum for the $E_2$-term. These can be used to determine the
roots of Bernstein-Sato polynomials supported at the origin, except for rather
special cases.
