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We investigate the distribution of zeros of the Lerch transcendent function We find an upper and lower estimates of zeros of the function Φ(q,s,a) in any rectangle {s : σ1 < Re s < σ2 ≤ 1.73…, 0 < Im s ≤ T}. Further we are interested in a computer calculations concerning the zeros of Φ(q,s,α) in {s : Re s > 1, 0 < Im s ≤ 1000}.
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