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

Eisenstein series