Geodesic
On~the zeros of the Davenport--Heilbronn function lying on the critical line
Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 311-324

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It is proved that at least $T(\log T)^{0,5-\varepsilon}$ zeros of the Davenport-Heilbronn function lie on the segment $(0,T]$ of the critical line. 
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     author = {A. A. Karatsuba},
     title = {On~the zeros of the {Davenport--Heilbronn} function lying on the critical line},
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A. A. Karatsuba. On~the zeros of the Davenport--Heilbronn function lying on the critical line. Izvestiya. Mathematics , Tome 36 (1991) no. 2, pp. 311-324. http://geodesic.mathdoc.fr/item/IM2_1991_36_2_a4/