The Generalized Riemann Hypothesis from zeros of a single L-function

For each primitive Dirichlet character $\chi$, a hypothesis ${\mathrm{G}\mathrm{R}\mathrm{H}}^{†}[\chi ]$ is formulated in terms of zeros of the associated $L$-function $L(s,\chi )$. It is shown that for any such character, ${\mathrm{G}\mathrm{R}\mathrm{H}}^{†}[\chi ]$ is equivalent to the Generalized Riemann Hypothesis.