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Using the relations between rational functions and Eisenstein series, as wellas the inferences for cotangent sums and period polynomials, we work out aprecise description for Eisenstein series whose $L$-series vanish at certaincritical values. This is possible for small weights compared to the level ofthe Eisenstein series. For large weights we give a partial result and determinesubspaces with simultaneous vanishing properties.

L-series of Eisenstein series vanishing at critical values