Yes, Speiser's theorem is an if and only if.

See Theorem 1 and "Corollary to Theorem 1" in Levinson and Montgomery's [Zeros of the derivatives of the Riemann Zeta-function][1]. Acta Math. 133 (1974), 49–65. 


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Edit:  An English language explication of Speiser's proof can be found in Arias-de-Reyna's [X-ray of Riemann Zeta Function][2]

  [1]: https://doi.org/10.1007/BF02392141
[2]: https://arxiv.org/abs/math/0309433