The Grand Riemann Hypothesis is the assertion that the nontrivial zeros of all automorphic $L$-functions lie on the critical line.
The Modified Grand Riemann Hypothesis is the assertion that the nontrivial zeros of all automorphic $L$-functions lie on the critical line or the real line.
It is widely believed that all global $L$-functions are automorphic $L$-functions. Presumably this also coincides with the Selberg class.
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