L(s) = 1 | + (−0.659 − 0.751i)2-s + (−0.602 + 0.798i)3-s + (−0.128 + 0.991i)4-s + (0.997 − 0.0738i)5-s + (0.997 − 0.0738i)6-s + (−0.993 − 0.110i)7-s + (0.830 − 0.557i)8-s + (−0.273 − 0.961i)9-s + (−0.713 − 0.700i)10-s + (0.989 − 0.147i)11-s + (−0.713 − 0.700i)12-s + (0.0922 − 0.995i)13-s + (0.572 + 0.819i)14-s + (−0.542 + 0.840i)15-s + (−0.966 − 0.255i)16-s + (−0.763 + 0.645i)17-s + ⋯ |
L(s) = 1 | + (−0.659 − 0.751i)2-s + (−0.602 + 0.798i)3-s + (−0.128 + 0.991i)4-s + (0.997 − 0.0738i)5-s + (0.997 − 0.0738i)6-s + (−0.993 − 0.110i)7-s + (0.830 − 0.557i)8-s + (−0.273 − 0.961i)9-s + (−0.713 − 0.700i)10-s + (0.989 − 0.147i)11-s + (−0.713 − 0.700i)12-s + (0.0922 − 0.995i)13-s + (0.572 + 0.819i)14-s + (−0.542 + 0.840i)15-s + (−0.966 − 0.255i)16-s + (−0.763 + 0.645i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4780559432 + 0.3717192296i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4780559432 + 0.3717192296i\) |
\(L(1)\) |
\(\approx\) |
\(0.6251782084 + 0.01609587511i\) |
\(L(1)\) |
\(\approx\) |
\(0.6251782084 + 0.01609587511i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.659 - 0.751i)T \) |
| 3 | \( 1 + (-0.602 + 0.798i)T \) |
| 5 | \( 1 + (0.997 - 0.0738i)T \) |
| 7 | \( 1 + (-0.993 - 0.110i)T \) |
| 11 | \( 1 + (0.989 - 0.147i)T \) |
| 13 | \( 1 + (0.0922 - 0.995i)T \) |
| 17 | \( 1 + (-0.763 + 0.645i)T \) |
| 19 | \( 1 + (-0.273 + 0.961i)T \) |
| 23 | \( 1 + (-0.966 + 0.255i)T \) |
| 29 | \( 1 + (-0.993 + 0.110i)T \) |
| 31 | \( 1 + (-0.343 + 0.938i)T \) |
| 37 | \( 1 + (-0.945 - 0.326i)T \) |
| 41 | \( 1 + (-0.273 + 0.961i)T \) |
| 43 | \( 1 + (-0.201 - 0.979i)T \) |
| 47 | \( 1 + (0.572 + 0.819i)T \) |
| 53 | \( 1 + (0.631 - 0.775i)T \) |
| 59 | \( 1 + (0.956 - 0.291i)T \) |
| 61 | \( 1 + (0.572 + 0.819i)T \) |
| 67 | \( 1 + (0.309 + 0.951i)T \) |
| 71 | \( 1 + (0.830 - 0.557i)T \) |
| 73 | \( 1 + (0.572 - 0.819i)T \) |
| 79 | \( 1 + (0.165 + 0.986i)T \) |
| 83 | \( 1 + (-0.602 + 0.798i)T \) |
| 89 | \( 1 + (-0.982 - 0.183i)T \) |
| 97 | \( 1 + (0.739 + 0.673i)T \) |
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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
−21.88731587825540174706997776571, −20.312620556494478280420589448887, −19.61278336579549908180787877339, −18.785332938355106573860402375663, −18.3148023811153611431573841425, −17.389526065352358076497764495696, −16.91595664648703487775533474351, −16.28317211704845497096905948337, −15.31382501160534107310151843666, −14.134683317804687453309904269966, −13.67183380553512989986967286031, −12.909327321932661639815001485591, −11.76109136656188296685202866221, −10.99277540659371715054512288513, −9.935895711884716211859522516065, −9.26919860370516989901416751416, −8.63609722152725770377005736955, −7.11263627155270983578282355738, −6.7870752411370508441627438687, −6.139696614069311977764692033805, −5.36664702782137524979003124883, −4.21505276593775164557541991906, −2.352856326271171315579277074584, −1.74234001863452326060126108701, −0.387341756841946158733578034346,
1.085259942418767902520184714391, 2.21797075933674710151549302062, 3.52494631403299396904610180635, 3.89587721261068062792218849201, 5.336939337635831390375241237, 6.17174487233522447378594927562, 6.92126813915716944871152877528, 8.457364256353162187913635082219, 9.13010940526912926801189763379, 9.93178132128829355982184760041, 10.33035166763449398553960549541, 11.11427924166084608120470106385, 12.22329797428952786295258900779, 12.74777078906877249983225341027, 13.65737394617523072101747293891, 14.70818076646083553639396826668, 15.83369970777200206066699435766, 16.53654840647450462703029128622, 17.17283120905391555758963251810, 17.728682375490193332369892999909, 18.534825207313511092985964652, 19.63667734919543013087774052296, 20.191667783489616751764316331624, 21.00105948724180627735248217372, 21.77476396136438120412843337132