# Fungrim entry: 6ae250

$\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{12}\!\left(\tau\right) = \left\{\frac{i \,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, a\right)}{\,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, 1 - a\right)}\right\}\; \text{ where } a = \frac{1}{2} + \frac{21 \sqrt{10} i}{100}$
TeX:
\mathop{\operatorname{zeros}\,}\limits_{\tau \in \mathcal{F}} E_{12}\!\left(\tau\right) = \left\{\frac{i \,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, a\right)}{\,{}_2F_1\!\left(\frac{1}{6}, \frac{5}{6}, 1, 1 - a\right)}\right\}\; \text{ where } a = \frac{1}{2} + \frac{21 \sqrt{10} i}{100}
Definitions:
Fungrim symbol Notation Short description
Zeros$\mathop{\operatorname{zeros}\,}\limits_{x \in S} f(x)$ Zeros (roots) of function
EisensteinE$E_{k}\!\left(\tau\right)$ Normalized Eisenstein series
ModularGroupFundamentalDomain$\mathcal{F}$ Fundamental domain for action of the modular group
ConstI$i$ Imaginary unit
Hypergeometric2F1$\,{}_2F_1\!\left(a, b, c, z\right)$ Gauss hypergeometric function
Sqrt$\sqrt{z}$ Principal square root
Source code for this entry:
Entry(ID("6ae250"),
Formula(Equal(Zeros(EisensteinE(12, tau), ForElement(tau, ModularGroupFundamentalDomain)), Where(Set(Div(Mul(ConstI, Hypergeometric2F1(Div(1, 6), Div(5, 6), 1, a)), Hypergeometric2F1(Div(1, 6), Div(5, 6), 1, Sub(1, a)))), Equal(a, Add(Div(1, 2), Div(Mul(Mul(21, Sqrt(10)), ConstI), 100)))))))

## Topics using this entry

Copyright (C) Fredrik Johansson and contributors. Fungrim is provided under the MIT license. The source code is on GitHub.

2021-03-15 19:12:00.328586 UTC