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We show that the analytic continuations of Helson zeta functions $ \zeta_\chi(s)= \sum_1^{\infty}\chi(n)n^{-s} $ can have essentially arbitrary poles andzeroes in the strip $ 21/40 < \Re s < 1 $ (unconditionally), and in the wholecritical strip $ 1/2 < \Re s <1 $ under Riemann Hypothesis for the function $\chi $ taking values in cubic roots of unity. If the sets are symmetric withrespect to the real axis, the same can be achieved with $ \chi $ taking values$ \pm 1 $.

Helson zeta functions for characters with finitely many values