Modular lambda function: Difference between revisions

Revision as of 11:18, 12 May 2021

Revision as of 14:25, 12 May 2021

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:<math>\lambda^*(1) = \frac{1}{\sqrt{2}}</math>

:<math>\lambda^*(5) = \sin[\frac{1}{2}\arcsin\left(\left(\tfrac{\sqrt{5}-1}{2}\right)^3\right)]</math>

:<math>\lambda^*(9) = \frac{1}{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})</math>

:<math>\lambda^*(13) = \sin[\frac{1}{2}\arcsin\left(\left(\tfrac{5\sqrt{13}-3}{2}\right)^3\right18)]</math>

:<math>\lambda^*(17) = \sin\{\frac{1}{2}\arcsin[\frac{1}{64}(5+\sqrt{17}-\sqrt{10\sqrt{17}+26})^3]\}</math>

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:<math>\lambda^*(33) = \sin\{\frac{1}{2}\arcsin[(10-3\sqrt{11})(2-\sqrt{3})^3]\}</math>

:<math>\lambda^*(37) = \sin[\{\frac{1}{2}\arcsin\left(\left[(\sqrt{37}-6\right)^3\right)]\}</math>

:<math>\lambda^*(45) = \sin\{\frac{1}{2}\arcsin[(4-\sqrt{15})^2(\sqrt{5}-2)^3]\}</math>

:<math>\lambda^*(49) = \sin\{\frac{1}{24}(8+3\arcsin[sqrt{7})(5-\fracsqrt{17}-\sqrt[4]{828})(4\sqrt{714}+9-3\sqrt{2}-\sqrt[8]{28}\sqrt{5-\sqrt{7}+21})^3]\}</math>

:<math>\lambda^*(57) = \sin\{\frac{1}{2}\arcsin[(170-39\sqrt{19})(2-\sqrt{3})^3]\}</math>

:<math>\lambda^*(73) = \sin\{\frac{1}{2}\arcsin[\frac{1}{64}(45+5\sqrt{73}-3\sqrt{50\sqrt{73}+426})^3]\}</math>

Lambda-star-values of integer numbers of 4n-2-type:

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:<math>\lambda^*(58) = (13\sqrt{58}-99)(\sqrt{2}-1)^6</math>

:<math>\lambda^*(6270) = \tan\{\frac{1}{2}\arctan[\frac{1}{512}(\sqrt{\sqrt{380\sqrt{2}+529}+10\sqrt{2}+195}-\sqrt{\sqrt{380\sqrt{2}+529}+10)^4(\sqrt{2}+11}-1)^6]\}</math>

:<math>\lambda^*(6678) = \tan\{\frac{1}{42}\arcsinarctan[\frac{1}{11532}(135\sqrt{3}+\sqrt{1113}-18)^2(\sqrt{1142\sqrt{3326}-5930}5)^2]\}</math>

:<math>\lambda^*(7082) = \tan\{\frac{1}{24}\arctanarcsin[(\sqrtfrac{51}{4761}-2)^4(8\sqrt{241}-151)^62]\}</math>

Lambda-star-values of integer numbers of 4n-1-type: