  We compute the one-level density of the non-trivial zeros of the Riemann
zeta-function weighted by $|\zeta(\frac12+it)|^{2k}$ for $k=1$ and, for test
functions with Fourier support in $(-\frac12,\frac12)$, for $k=2$. As a
consequence, for $k=1,2$, we deduce under the Riemann hypothesis that $T(\log
T)^{1-k^2+o(1)}$ non-trivial zeros of $\zeta$, of imaginary parts up to $T$,
are such that $\zeta$ attains a value of size $(\log T)^{k+o(1)}$ at a point
which is within $O(1/\log T)$ from the zero.
