Infinitely many GL(3) cusp forms have both central L-values L(1/2, pi x chi) and L(1/2, pi x sigma) non-zero simultaneously.
Average values of modular L -series via the relative trace formula
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Relative Trace Formula And Simultaneous Nonvanishing for GL_3 x GL_2 and GL_3 x GL_1 L-functions
Infinitely many GL(3) cusp forms have both central L-values L(1/2, pi x chi) and L(1/2, pi x sigma) non-zero simultaneously.