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===Special Values===
Lambda-star-values of integer numbers of 4n-3-type:
:<math>\lambda^*(1) = \frac{1}{\sqrt{2}}</math>
:<math>\lambda^*(25) = \sqrtsin[\frac{1}{2}\arcsin(\sqrt{5}-12)]</math>
:<math>\lambda^*(39) = \frac{1}{2}(\sqrt{2}3}-1)(\sqrt{32}-1\sqrt[4]{3})</math>
:<math>\lambda^*(413) = \sin[\frac{1}{2}\arcsin(5\sqrt{213}-118)^2]</math>
:<math>\lambda^*(517) = \sin\{\frac{1}{2}\sqrtarcsin[\frac{21}{64}(5+\sqrt{217}-\sqrt{5}-2}-10\sqrt{517}+126})^3]\}</math>
:<math>\lambda^*(621) = (\sin\{\frac{1}{2}\arcsin[(8-3\sqrt{37})(2\sqrt{37}-3\sqrt{23})]\}</math>
:<math>\lambda^*(725) = \frac{1}{4\sqrt{2}}(\sqrt{5}-2)(3-2\sqrt[4]{75})</math>
Lambda-star-values of integer numbers of 4n-2-type:
:<math>\lambda^*(8) = (\sqrt{2}+1)(\sqrt{\sqrt{2}+1}-\sqrt{2})^2</math> ▼
:<math>\lambda^*(92) = \frac{1}sqrt{2}(\sqrt{3}-1)(\sqrt{2}-\sqrt[4]{3})</math>
▲:<math>\lambda^*( 86) = ( 2-\sqrt{ 23} +1)(\sqrt{ \sqrt{2}+13}-\sqrt{2}) ^2</math>
:<math>\lambda^*(10) = (\sqrt{10}-3)(\sqrt{2}-1)^2</math>
:<math>\lambda^*(1114) = \tan\{\frac{1}{8\sqrt{2}}(\sqrt{11}+3)(arctan[\frac{1}{38}(2\sqrt[3]{6\sqrt{32}+2\sqrt{11}}1-\frac{1}{3}\sqrt[3]{64\sqrt{3}-2\sqrt{11}}+\frac{15}{)^3}]\sqrt{11}-1)^4</math>
:<math>\lambda^*(1218) = (\sqrt{32}-\sqrt{2}1)^3(2(-\sqrt{23}-1)^2</math>
:<math>\lambda^*(1322) = \frac{1}{2\sqrt{2}}[(5+10-3\sqrt{1311})(3\sqrt{5\sqrt{1311}-18}-5+7\sqrt{132}])</math>
Lambda-star-values of integer numbers of 4n-1-type:
:<math>\lambda^*(14) = [2\sqrt{2}+2-(\sqrt{2}+1)^2\sqrt{4\sqrt{2}-5}][(\sqrt{2}+1)^2\sqrt{4\sqrt{2}-5}-\sqrt{8\sqrt{2}+10}]</math> ▼
:<math>\lambda^*(153) = \frac{1}{82\sqrt{2}}(3-\sqrt{5})(\sqrt{5}-\sqrt{3})(2-\sqrt{3}1)</math>
:<math>\lambda^*(167) = (\frac{1}{4\sqrt{2}+1)^2}(3-\sqrt[4]{27}-1)^4</math>
:<math>\lambda^*(1711) = \frac{1}{8\sqrt{2}}(\sqrt{11}+3)(\sqrtfrac{171}+11{3}-\sqrt[3]{56\sqrt{3}+2\sqrt{1711}}-\frac{1}{3}\sqrt[3]{26\sqrt{3}-2\sqrt{1711}}+2\frac{1}-4{3}\sqrt{11}-1)^24</math>
:<math>\lambda^*(1815) = (\frac{1}{8\sqrt{2}}(3-1\sqrt{5})^(\sqrt{5}-\sqrt{3})(2-\sqrt{3})^2</math>
:<math>\lambda^*(19) = \frac{1}{8\sqrt{2}}(3\sqrt{19}+13)[\frac{1}{6}(\sqrt{19}-2+\sqrt{3})\sqrt[3]{3\sqrt{3}-\sqrt{19}}-\frac{1}{6}(\sqrt{19}-2-\sqrt{3})\sqrt[3]{3\sqrt{3}+\sqrt{19}}-\frac{1}{3}(5-\sqrt{19})]^4</math>
:<math>\lambda^*(2023) = (\frac{1}{16\sqrt{102}}-3)(5+\sqrt{523}+2)[\frac{1}{6}(\sqrt{23}-+1)(\sqrt[3]{100-12\sqrt{569}}-\frac{1}{6}(\sqrt{3}-1)^\sqrt[3]{100+12\sqrt{69}}+\frac{2}{3}]^4</math>
Lambda-star-values of integer numbers of 4n-type:
:<math>\lambda^*(21) = \frac{1}{4\sqrt{2}}(\sqrt{7}-\sqrt{3})[(\sqrt{3}+1)\sqrt{2\sqrt{7}-4}-4+\sqrt{7}-\sqrt{3}]</math>
:<math>\lambda^*(224) = (10-3\sqrt{112}-1)(3\sqrt{11}-7\sqrt{^2})</math>
:<math>\lambda^*(238) = \frac{1}{16\sqrt{2}}(5+\sqrt{23})tan[\frac{1}{62}(\sqrt{3}+1)\sqrt[3]{100-12\sqrt{69}}-\frac{1}{6}arcsin(\sqrt{32}-1)\sqrt[3]{100+12\sqrt{69}}+\frac{^2}{3}]^4</math>
:<math>\lambda^*(2412) = (2+\sqrt{3})^2(\sqrt{3}+-\sqrt{2})[\sqrt{\sqrt{3}+\sqrt{^2}}-(\sqrt{32}-1)(\sqrt{2}+1)]^2</math>
:<math>\lambda^*(2516) = \frac{1}{(\sqrt{2}}(\sqrt{5}-+1)^2)(3-2\sqrt[4]{52}-1)^4</math>
:<math>\lambda^*(20) = \tan[\frac{1}{4}\arcsin(\sqrt{5}-2)]^2</math>
▲:<math>\lambda^*( 1424) = [2\ sqrttan\{ 2}+2-(\ sqrtfrac{ 21} +1)^2\sqrt{4\sqrt{2} -5}]\arcsin[( 2-\sqrt{ 23} +1) ^2(\sqrt{ 4\sqrt{2}-53}- \sqrt{8\sqrt{2} +10)]\} ]^2</math>
Lambda-star-values of rational fractions:
:<math>\lambda^*(\frac{3}{5}) = \frac{1}{8\sqrt{2}}(3+\sqrt{5})(\sqrt{5}-\sqrt{3})(2+\sqrt{3})</math>
:<math>\lambda^*(\frac{4}{5}) = (\sqrttan[\frac{10\pi}+3)(\sqrt{54}+2)(-\sqrtfrac{21}+1){4}\arcsin(\sqrt{\sqrt{5}-1}-12)]^2</math>
== Other appearances ==
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