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Fungrim entry: 6ae250

zerosτFE12 ⁣(τ)={i2F1 ⁣(16,56,1,a)2F1 ⁣(16,56,1,1a)}   where a=12+2110i100\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
ZeroszerosP(x)f ⁣(x)\mathop{\operatorname{zeros}\,}\limits_{P\left(x\right)} f\!\left(x\right) Zeros (roots) of function
EisensteinEEk ⁣(τ)E_{k}\!\left(\tau\right) Normalized Eisenstein series
ModularGroupFundamentalDomainF\mathcal{F} Fundamental domain for action of the modular group
ConstIii Imaginary unit
Hypergeometric2F12F1 ⁣(a,b,c,z)\,{}_2F_1\!\left(a, b, c, z\right) Gauss hypergeometric function
Sqrtz\sqrt{z} Principal square root
Source code for this entry:
Entry(ID("6ae250"),
    Formula(Equal(Zeros(EisensteinE(12, tau), tau, Element(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)))))))

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2019-09-16 21:17:18.797188 UTC