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We locate all but $O(\sqrt{k\log{k}})$ zeroes of the half integral weightEisenstein series $E_\infty(z,k)$ of $\Gamma_0(4)$ for $k$ sufficiently large.To do this, we relate $E_\infty(z,k)$ to $\Gamma_0(4)$'s other Eisensteinseries, $E_0(z,k)$ and $E_\frac{1}{2}(z,k)$, which are easier to study in theregions of which zeroes occur. We will use trigonometric approximations of$E_0(z,k)$ and $E_\frac{1}{2}(z,k)$ in order to locate the zeroes.

On the Zeroes of Half-Integral Weight Eisenstein Series on $\Gamma_0(4)$