L(s) = 1 | + (−0.342 − 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.642 + 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (0.866 + 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.642 + 0.766i)33-s + (0.939 + 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
L(s) = 1 | + (−0.342 − 0.939i)3-s + (0.984 + 0.173i)7-s + (−0.766 + 0.642i)9-s + (−0.5 − 0.866i)11-s + (−0.642 + 0.766i)13-s + (0.642 + 0.766i)17-s + (−0.939 + 0.342i)19-s + (−0.173 − 0.984i)21-s + (−0.866 − 0.5i)23-s + (0.866 + 0.5i)27-s + (0.5 + 0.866i)29-s + 31-s + (−0.642 + 0.766i)33-s + (0.939 + 0.342i)39-s + (0.766 + 0.642i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0634i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.247329518 + 0.03960480601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.247329518 + 0.03960480601i\) |
\(L(1)\) |
\(\approx\) |
\(0.9540754776 - 0.1535881027i\) |
\(L(1)\) |
\(\approx\) |
\(0.9540754776 - 0.1535881027i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 \) |
good | 3 | \( 1 + (-0.342 - 0.939i)T \) |
| 7 | \( 1 + (0.984 + 0.173i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.642 + 0.766i)T \) |
| 17 | \( 1 + (0.642 + 0.766i)T \) |
| 19 | \( 1 + (-0.939 + 0.342i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 + T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 - iT \) |
| 47 | \( 1 + (-0.866 - 0.5i)T \) |
| 53 | \( 1 + (0.984 - 0.173i)T \) |
| 59 | \( 1 + (0.173 + 0.984i)T \) |
| 61 | \( 1 + (0.766 + 0.642i)T \) |
| 67 | \( 1 + (-0.984 - 0.173i)T \) |
| 71 | \( 1 + (0.939 - 0.342i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + (-0.173 + 0.984i)T \) |
| 83 | \( 1 + (0.642 + 0.766i)T \) |
| 89 | \( 1 + (0.173 + 0.984i)T \) |
| 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
−20.84016882050987348023433285497, −20.11198423364651664907606608312, −19.30757593041762571120839697997, −18.02251094374396338650011535010, −17.61716237366926815310974718028, −17.0520273727860885449622397305, −15.998005194282584334894018880920, −15.40194703405603680948297309252, −14.71542652281653472045755108162, −14.10272621527831595941577132647, −12.98652873421928963975421270456, −12.02213103640240431275656148961, −11.51050460656179741205671345038, −10.472213141634394137464017392640, −10.07537957316679889445343976370, −9.24458576964191595150744896670, −8.10280587196472149058946608835, −7.63431846506937467529486632289, −6.40487004470667209781995984617, −5.4065691817570232354052640990, −4.77891188152245344276661207258, −4.19438241165599430899365570621, −2.965728473883945555182588016634, −2.09789690941858909117359130627, −0.59073636714912533871741314426,
0.963878217054549540250672393075, 1.94366541005629782850514988433, 2.64136714138803090416125497659, 4.01063742457112351419931979303, 4.99633387347823200740389081635, 5.77932405172074527949656527473, 6.521394616689693076703555473684, 7.45975854603066953466408168309, 8.34181446242416829081290773069, 8.57546103847899252391406899791, 10.131858329784192926436072437224, 10.79984413360759706311989794206, 11.6566364173340270586764325773, 12.20745891078863046611609874375, 12.972132297533040941780069535644, 13.986227130013512686246385256745, 14.35967884015780515997415794887, 15.26299901318107739774348142757, 16.5963132320468839346584608324, 16.7826234118326455391393763843, 17.88586853138196038451929647038, 18.29703117165099462015847534349, 19.222910474141518290065122155143, 19.561957180700835029673308046534, 20.81029980341337683606555634790