All complex conjugate zeros of θ(q,x) with Re(x)≥0 lie in 1<|x|<5 for q∈(0,1), none exist for q≤0.6687..., and those with Re(x)<0 lie in |x|<49.8.
Mortenson, On the dual nature of partial theta func tions and Appell-Lerch sums, Adv
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On the location of the complex conjugate zeros of the partial theta function
All complex conjugate zeros of θ(q,x) with Re(x)≥0 lie in 1<|x|<5 for q∈(0,1), none exist for q≤0.6687..., and those with Re(x)<0 lie in |x|<49.8.