A correction is brought to the opinion expressed in a previous note published in this journal that the off critical line points indicated as being non trivial zeros of Davenport and Heilbronn function are affected of approximation errors and illustrations are presented which enforce the conclusion that they are true zeros. It is shown also that linear combinations of L-functions satisfying the same Riemann-type of functional equation do not offer counterexamples to RH, contrary to a largely accepted position.

The existence of non trivial zeros off the critical line for a function obtained by analytic continuation of a particular Dirichlet series is studied. Our findings are in contradiction with some results of computations which were present in the field for a long time. In the first part of the note we illustrate how the approximation errors may have had as effect inexact conclusions and in the second part we prove rigorously our point of view. AMS subject classification: 30C35, 11M26Examples of functions obtained by analytic continuation of Dirichlet series, which have zeros off the line Re s  1/2 are important for the purpose of circumscribing the field where the Riemann Hypothesis might be true. Such a candidate is attributed by some mathematicians to Davenport and Heilbronn (1936) (see [1]) and by others to Titchmarsh [5] (see [4]). In [6], R. C. Vaughan provided an elementary clear presentation of the respective example using the robust theory from [3]. He starts with that Dirichlet character  modulo 5 for which 2  i. For this characterRepeating the same computation for the character  it can be easily found that   . With the notation used in Mathematica implementation of Dirichlet L-functions, L5, 2, s   n1  nn s and L5, 3, s   n1  nn s for Re s  1, it is known that L5, 2, s and L5, 3, s have analytic continuations to the whole complex plane and they are entire functions. Moreover, by corollary 10.9 of [3], they verify the functional equations:(3) L5, 2, s  2 s  s1 5 1/2s 1  scos s 2 L5, 3, 1  s (4) L5, 3, s  2 s  s1 5 1/2s 1  scos s 2 L5, 2, 1  s

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