The derivative E6' has exactly one or two zeros per Gamma0(2) fundamental domain and zero or one per SL(2,Z) domain, with the zeros accumulating on three explicit curves.
Akhiezer; Elements of the theory of elliptic functions
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On the distribution of critical points of the Eisenstein series $E_6$ and monodromy interpretation
The derivative E6' has exactly one or two zeros per Gamma0(2) fundamental domain and zero or one per SL(2,Z) domain, with the zeros accumulating on three explicit curves.