A detailed note on the zeros of Eisenstein series for Gamma₀^* (5) and Gamma₀^* (7)

The present paper provides the details omitted from the more concise study "On the zeros of Eisenstein series for $\Gamma_0^* (5)$ and $\Gamma_0^* (7)$." We locate almost all of the zeros of the Eisenstein series associated with the Fricke groups of level 5 and 7 in their fundamental domains by applying and extending the method of F. K. C. Rankin and H. P. F. Swinnerton-Dyer (1970). We also use the arguments of some terms of the Eisenstein series in order to improve existing error bounds.