Where not to find the spectrum of the partial theta function

The spectrum of Ramanujan’s partial theta function $\theta (q,x):={\sum }_{j=0}^{\mathrm{\infty }}{q}^{j(j+1)\mathrm{/}2}{x}^j$, $q\in {\mathbb{D}}_1$ (the unit disk centered at the origin), $x\in \mathbb{C}$, is the set of values of the parameter $q$ for which $\theta (q,\mathrm{.})$ has a multiple zero. We show that there is no spectral value in the set $\mathbb{S}\cup {\mathbb{D}}_{c_0}$, $c_0=0.2078750206\dots$, where $\mathbb{S}$ is the sector $\{0<\mathrm{\mid }z\mathrm{\mid }<0.6,\mathrm{a}\mathrm{r}\mathrm{g}(z)\in [\pi \mathrm{/}4,7\pi \mathrm{/}4]\}$. There is a single spectral value in the set $\mathbb{S}\cup {\mathbb{D}}_{0.31}$ which equals $0.309249\dots$. For $q\in \mathbb{S}\cup {\mathbb{D}}_{c_0}$, the moduli of the zeros of $\theta$ are separated by the negative half-integer powers of $\mathrm{\mid }q\mathrm{\mid }$.