L(s) = 1 | + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (0.438 + 0.898i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.939 − 0.342i)23-s + (−0.882 + 0.469i)25-s + (−0.309 + 0.951i)26-s + ⋯ |
L(s) = 1 | + (0.719 + 0.694i)2-s + (0.0348 + 0.999i)4-s + (0.241 + 0.970i)5-s + (−0.615 + 0.788i)7-s + (−0.669 + 0.743i)8-s + (−0.5 + 0.866i)10-s + (0.438 + 0.898i)13-s + (−0.990 + 0.139i)14-s + (−0.997 + 0.0697i)16-s + (−0.913 + 0.406i)17-s + (0.669 − 0.743i)19-s + (−0.961 + 0.275i)20-s + (0.939 − 0.342i)23-s + (−0.882 + 0.469i)25-s + (−0.309 + 0.951i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 297 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.828 - 0.559i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.5824998027 + 1.902608320i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.5824998027 + 1.902608320i\) |
\(L(1)\) |
\(\approx\) |
\(0.8701763317 + 1.065322507i\) |
\(L(1)\) |
\(\approx\) |
\(0.8701763317 + 1.065322507i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.719 + 0.694i)T \) |
| 5 | \( 1 + (0.241 + 0.970i)T \) |
| 7 | \( 1 + (-0.615 + 0.788i)T \) |
| 13 | \( 1 + (0.438 + 0.898i)T \) |
| 17 | \( 1 + (-0.913 + 0.406i)T \) |
| 19 | \( 1 + (0.669 - 0.743i)T \) |
| 23 | \( 1 + (0.939 - 0.342i)T \) |
| 29 | \( 1 + (-0.990 - 0.139i)T \) |
| 31 | \( 1 + (0.559 - 0.829i)T \) |
| 37 | \( 1 + (0.669 + 0.743i)T \) |
| 41 | \( 1 + (-0.990 + 0.139i)T \) |
| 43 | \( 1 + (0.766 - 0.642i)T \) |
| 47 | \( 1 + (-0.0348 + 0.999i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.848 + 0.529i)T \) |
| 61 | \( 1 + (0.559 + 0.829i)T \) |
| 67 | \( 1 + (0.173 - 0.984i)T \) |
| 71 | \( 1 + (-0.913 + 0.406i)T \) |
| 73 | \( 1 + (-0.978 + 0.207i)T \) |
| 79 | \( 1 + (-0.719 - 0.694i)T \) |
| 83 | \( 1 + (-0.438 + 0.898i)T \) |
| 89 | \( 1 + (0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.241 + 0.970i)T \) |
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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
−24.64192294447627230486768924209, −23.46473728537194894334270034275, −22.91245940536854825171838210869, −21.956586565322772731508923744505, −20.85483031471163731072010380916, −20.26527439115191267923195460858, −19.66054045388139425576179415452, −18.44435263567655048194095968299, −17.35954256929732151480990854280, −16.24170877118143737319290729788, −15.488479635789474730825176642277, −14.16557088177191885796375118615, −13.22749251271265077809087659485, −12.87616652356390665542448165814, −11.69412142067448681625449816377, −10.62947387677001763777454373998, −9.74893905237020779570397317231, −8.8188578699878670728829003451, −7.30075380308497265634868001963, −6.020499041490539598626169629775, −5.109210922540733955430175048618, −4.04009254360195226363702674303, −3.03199815880009193116693481439, −1.47436158564015122241487730113, −0.45707494260898525712209614722,
2.2826455010276898086543393087, 3.16383988754347964589908808688, 4.3635360554784234247599585889, 5.72344264240585557784291909810, 6.5029186756632909742728303988, 7.23184026034386235658844441074, 8.66053735892187350144708034165, 9.520275855780952232492774588304, 11.08045775815496206439680115886, 11.79809012484137327696417603175, 13.11552806728770416079642752066, 13.6808821743362563957419412559, 14.90949878729398404202356494462, 15.37550906909228041624056509498, 16.394495158657325340323404458868, 17.43574303823464683066522815785, 18.41131261315949493070937030359, 19.16743959889895336927120539482, 20.59351738976617447055830683046, 21.657448851267845127823073116249, 22.205194527920473097163368985203, 22.86558132211425692924991504807, 23.97323078334814308550739102703, 24.75544931231149480827355229756, 25.8716054352566296576287021311