L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + 19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.5 − 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)7-s − 8-s + 10-s + (0.5 − 0.866i)11-s + (−0.5 − 0.866i)13-s + (0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s − 17-s + 19-s + (0.5 − 0.866i)20-s + (−0.5 − 0.866i)22-s + (0.5 + 0.866i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.113946943 - 0.5194419906i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.113946943 - 0.5194419906i\) |
\(L(1)\) |
\(\approx\) |
\(1.136275918 - 0.4135706123i\) |
\(L(1)\) |
\(\approx\) |
\(1.136275918 - 0.4135706123i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + (0.5 - 0.866i)T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 + 0.866i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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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
−48.3575107148706247398265274126, −45.96161983593313488222891071970, −44.35691248613978483473807819478, −43.28558880343466646432456708351, −41.693019157648402269033143994997, −40.3943285623758754227281525746, −39.07364690643271231694332487226, −36.48187828065801756344300334031, −35.48489755956410935564836049716, −33.412433807129446974963822755490, −32.573271591370362790797457866918, −30.90230473712494219306650729593, −28.9577216343121731590172682939, −26.763428341842772681883211365481, −25.21569578513310088903785578470, −23.83477567755688274260462694769, −22.1765179023783446952084486311, −20.29865916653771873113403507250, −17.47183618509653816175409815785, −16.2551781515321363983903806222, −14.08927896436677198005366627039, −12.586494690151173034212579462720, −9.262408964681987903208968444344, −6.89180300406244509022667855056, −4.57573576242485587483690886648,
2.90199460773728348151070380596, 5.911589958627904376506207333864, 9.56544293453670367607222319631, 11.407478947536363718469485584758, 13.38637373763353051188582047267, 15.105372647488006472027705459194, 18.1206180739956670326074406043, 19.55064984717247446010842616849, 21.6799089672262001832976919749, 22.55618331766574485452776535305, 24.80445956457406238489997737585, 26.9541568022766289467089005514, 28.821574701722030960999337413674, 29.99559371181156904706964137279, 31.55515371295916543880032766067, 33.04716874508529738151387747530, 34.97343033685754567017430433836, 37.27484517767191228753209201241, 38.02702866170852311461138362749, 39.68758273933563024241709715528, 41.265221376972713038042216016559, 42.19507464737363382501335827663, 44.781583927578735998719033134104, 45.95935080542520981794114449963, 47.55248127455483201705852602745