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We prove that 33% of $L (1/2+it_f, f)$ for $ t_f \\leqslant T$ do not vanish as $T \\rightarrow \\infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. 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We prove that 33% of $L (1/2+it_f, f)$ for $ t_f \\leqslant T$ do not vanish as $T \\rightarrow \\infty$. For comparison, it is known that the non-vanishing proportion is at least 25% for the central $L$-values $L (1/2, f)$. Further, 33% may be raised to 50% conditionally on the generalized Riemann hypothesis. 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