L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.984 − 0.173i)20-s + (0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (−0.173 − 0.984i)4-s + (0.342 − 0.939i)5-s + (−0.939 − 0.342i)7-s + (−0.866 − 0.5i)8-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)11-s + (0.984 − 0.173i)13-s + (−0.866 + 0.5i)14-s + (−0.939 + 0.342i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (−0.984 − 0.173i)20-s + (0.342 + 0.939i)22-s + (0.866 − 0.5i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 111 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.464 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6758416868 - 1.117881813i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6758416868 - 1.117881813i\) |
\(L(1)\) |
\(\approx\) |
\(1.019690603 - 0.8160879428i\) |
\(L(1)\) |
\(\approx\) |
\(1.019690603 - 0.8160879428i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 37 | \( 1 \) |
good | 2 | \( 1 + (0.642 - 0.766i)T \) |
| 5 | \( 1 + (0.342 - 0.939i)T \) |
| 7 | \( 1 + (-0.939 - 0.342i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 13 | \( 1 + (0.984 - 0.173i)T \) |
| 17 | \( 1 + (0.984 + 0.173i)T \) |
| 19 | \( 1 + (-0.642 - 0.766i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + iT \) |
| 41 | \( 1 + (0.173 + 0.984i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.939 - 0.342i)T \) |
| 59 | \( 1 + (-0.342 - 0.939i)T \) |
| 61 | \( 1 + (-0.984 + 0.173i)T \) |
| 67 | \( 1 + (0.939 + 0.342i)T \) |
| 71 | \( 1 + (-0.766 + 0.642i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.342 + 0.939i)T \) |
| 83 | \( 1 + (-0.173 + 0.984i)T \) |
| 89 | \( 1 + (0.342 + 0.939i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−29.82617570684338647833708311217, −29.14779340512032245041100666174, −27.4773569458315308147707437208, −26.28693844054171146455886839804, −25.70596694145591974416699330479, −24.84837534358111094761068622996, −23.30449207786046416897620664516, −22.89479821518036751579789273379, −21.64236396465830427188813040651, −21.03929270899617152787945708642, −19.034267498733526267059474190639, −18.3965369134507832831031311538, −16.94735632852782413267499263227, −15.98849099009566480936424421462, −15.040711617246579343399084442645, −13.86916328722721331400968401296, −13.122289438086927811958844732896, −11.75101685157455486233434518718, −10.392240155579355875046553775148, −8.93574568635934678186368050218, −7.607591558075890610450188653205, −6.28793213585456163900534447441, −5.720065349404640558043073827243, −3.7012332106416503053047628090, −2.79678588064394895149390756875,
1.171510158759399718803548661639, 2.86869010281102853206170058020, 4.28345031344728435540143927472, 5.42141263260524557403469017359, 6.69867796249142527564666990477, 8.69050066945790253419347318594, 9.81895863735654959659765434168, 10.73117706653836160688065845717, 12.40294371719106970852966125133, 12.90145285918337856123536836548, 13.87970103126866489643935955624, 15.34436545353739683660884928187, 16.3678651940966131107354830609, 17.73731665724978824004089666853, 19.02153827652765669468610672678, 20.05671415300269933410981578313, 20.81526232534080430167495549385, 21.72977993738743857960215153183, 23.18222470294665388526348509562, 23.47771288860960138597764303377, 25.01931427079757678550333277110, 25.85546660528239133774665157071, 27.51845560562923494335503640692, 28.45848936970113166157069802089, 28.98561392999534061936079882462