One-level density of zeros of the Riemann zeta function weighted by its special values

We consider the one-level density of zeros of the Riemann zeta function weighted by its special values on the line $\mathrm{R}\mathrm{e}(s)=\sigma$, where $\sigma$ is a fixed real number such that $1\mathrm{/}2\le \sigma \le 1$. We show that the density function for the Riemann zeta function changes if and only if $\sigma =1\mathrm{/}2$.