L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.374 − 0.927i)11-s + (−0.719 − 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (−0.374 − 0.927i)22-s + (0.615 − 0.788i)23-s + ⋯ |
L(s) = 1 | + (0.719 − 0.694i)2-s + (0.0348 − 0.999i)4-s + (−0.766 + 0.642i)5-s + (−0.615 − 0.788i)7-s + (−0.669 − 0.743i)8-s + (−0.104 + 0.994i)10-s + (0.374 − 0.927i)11-s + (−0.719 − 0.694i)13-s + (−0.990 − 0.139i)14-s + (−0.997 − 0.0697i)16-s + (0.978 − 0.207i)17-s + (0.913 − 0.406i)19-s + (0.615 + 0.788i)20-s + (−0.374 − 0.927i)22-s + (0.615 − 0.788i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 837 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.580 + 0.814i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.6200126913 - 1.203710589i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.6200126913 - 1.203710589i\) |
\(L(1)\) |
\(\approx\) |
\(0.8587881676 - 0.7713707986i\) |
\(L(1)\) |
\(\approx\) |
\(0.8587881676 - 0.7713707986i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.719 - 0.694i)T \) |
| 5 | \( 1 + (-0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.615 - 0.788i)T \) |
| 11 | \( 1 + (0.374 - 0.927i)T \) |
| 13 | \( 1 + (-0.719 - 0.694i)T \) |
| 17 | \( 1 + (0.978 - 0.207i)T \) |
| 19 | \( 1 + (0.913 - 0.406i)T \) |
| 23 | \( 1 + (0.615 - 0.788i)T \) |
| 29 | \( 1 + (0.719 - 0.694i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (-0.438 + 0.898i)T \) |
| 43 | \( 1 + (-0.241 - 0.970i)T \) |
| 47 | \( 1 + (-0.559 + 0.829i)T \) |
| 53 | \( 1 + (-0.309 + 0.951i)T \) |
| 59 | \( 1 + (0.241 - 0.970i)T \) |
| 61 | \( 1 + (-0.939 + 0.342i)T \) |
| 67 | \( 1 + (0.173 + 0.984i)T \) |
| 71 | \( 1 + (0.978 - 0.207i)T \) |
| 73 | \( 1 + (-0.978 - 0.207i)T \) |
| 79 | \( 1 + (-0.882 - 0.469i)T \) |
| 83 | \( 1 + (0.719 - 0.694i)T \) |
| 89 | \( 1 + (0.978 + 0.207i)T \) |
| 97 | \( 1 + (-0.374 + 0.927i)T \) |
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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
−22.71338637075098501625312550464, −21.76356956116139349078236008099, −21.07746532079914617113174864856, −20.12282334783248948306232625828, −19.3847482701917880039331365915, −18.447591807480060491250444499471, −17.353932665484002909269444714859, −16.61041427587690570972358136356, −15.99855120921200402912509753365, −15.18838841829383363579835380126, −14.61737491988260492990626655343, −13.568555392033287060367302291919, −12.531412690245167238692852432815, −12.14074113054657133845534640692, −11.58007404156474710957005030870, −9.843780154921176180984035104160, −9.129461600992026595550895132669, −8.201797273796572851098642838812, −7.315597615698995694770344305507, −6.63347769164464867211557612978, −5.36578079101899774361571794164, −4.883888053661052418149856358, −3.754671708715761636524836057709, −3.012577271632872832219989549884, −1.5797413519489663905905814653,
0.26717022380716008430914953740, 1.003764633741140177787716516800, 2.84629585018037757449034995001, 3.19648975746203665760055474965, 4.12307215184045046438185398138, 5.14309255339159055112620081595, 6.23185379309141769375049237262, 7.04067975070030210698987617497, 7.93405307810891697342879169254, 9.29898467075925149990186283250, 10.22158049759621062573064087871, 10.775191744324575531477337866945, 11.69502276067324517420187373618, 12.34038735501657491561931850973, 13.31884195785084380740464881075, 14.11934456984596746561253025351, 14.69752852842294519620132181486, 15.71281395791997566291624430570, 16.35181678257661069398504797523, 17.48664415548152516474330186845, 18.704259611388641003393423139630, 19.15288987542271909475539193957, 19.89785758126465497753825010536, 20.47758906088441784890824316682, 21.58725485684202612610919365027