L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 − 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (−0.766 + 0.642i)2-s + (0.173 − 0.984i)4-s + (0.173 − 0.984i)5-s + (0.5 + 0.866i)8-s + (0.5 + 0.866i)10-s + (−0.173 − 0.984i)11-s + (−0.173 + 0.984i)13-s + (−0.939 − 0.342i)16-s + (−0.5 − 0.866i)17-s + (0.5 − 0.866i)19-s + (−0.939 − 0.342i)20-s + (0.766 + 0.642i)22-s + (−0.766 − 0.642i)23-s + (−0.939 − 0.342i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.490 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5925859211 - 0.3462487061i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5925859211 - 0.3462487061i\) |
\(L(1)\) |
\(\approx\) |
\(0.7081269386 - 0.07992053504i\) |
\(L(1)\) |
\(\approx\) |
\(0.7081269386 - 0.07992053504i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 5 | \( 1 + (0.173 - 0.984i)T \) |
| 11 | \( 1 + (-0.173 - 0.984i)T \) |
| 13 | \( 1 + (-0.173 + 0.984i)T \) |
| 17 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (-0.173 - 0.984i)T \) |
| 31 | \( 1 + (-0.173 + 0.984i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.173 - 0.984i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (0.173 + 0.984i)T \) |
| 53 | \( 1 + (0.5 - 0.866i)T \) |
| 59 | \( 1 + (-0.939 + 0.342i)T \) |
| 61 | \( 1 + (-0.173 - 0.984i)T \) |
| 67 | \( 1 + (0.766 + 0.642i)T \) |
| 71 | \( 1 + (0.5 - 0.866i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (0.766 - 0.642i)T \) |
| 83 | \( 1 + (0.173 + 0.984i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.766 + 0.642i)T \) |
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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
−27.37475032869738554524197881665, −26.33386186130783415908912686447, −25.72331353295165534226327263005, −24.81236363563703924690600304936, −23.20537820803043723127875426084, −22.30866408870582156426817376990, −21.56710821488669911893118406685, −20.31193566225635847489899193651, −19.693815646950038425684552567531, −18.39357547775193712410417322357, −17.956801905501484792482913572088, −16.96405701206274567295087430736, −15.60653027680611061957273406594, −14.6987297604593132912768581109, −13.267055624279958870119739126896, −12.32402067919987137047476668377, −11.16258099614531859237062446671, −10.26186638341822213957001089275, −9.5817788630365743297879660724, −8.01711982183463288992819615168, −7.29461181498846211164580981224, −5.93709331056630743812250683389, −4.06389190818392793017235630134, −2.88579227453924662752734317377, −1.754704513425964360998335325,
0.697594061189452859001313377488, 2.26966804972312304482673448270, 4.43831982546875312310113420326, 5.49278453478733779421432255159, 6.60884680672832257355792381816, 7.86098937597596553521665258533, 8.91593878066542353233980425808, 9.501616225588817649267600630386, 10.92592180787891813865358142798, 11.9406259823837045896012135910, 13.50816408995621712696504187744, 14.1741559892971535171566010385, 15.756103351120371669623106452576, 16.25003836867843232729221924383, 17.180847584461595758276482011173, 18.15006011899561046604669296236, 19.1849350010354226669016324707, 20.07341472578763921434285220182, 21.04401596900599792633939282230, 22.24134643169335839263959282921, 23.68769294217498145635311751707, 24.26788387137284021254279353162, 24.94247655562223900950053060470, 26.14935560915976843331973197065, 26.85551993633735696623429415288