| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | − 2-s + 3-s + 4-s + 5-s − 6-s + 7-s − 8-s + 9-s − 10-s − 11-s + 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s + 19-s + 20-s + 21-s + 22-s + 23-s − 24-s + 25-s − 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1429 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.225997442\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.225997442\) |
| \(L(1)\) |
\(\approx\) |
\(1.386635330\) |
| \(L(1)\) |
\(\approx\) |
\(1.386635330\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1429 | \( 1 \) |
| good | 2 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
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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
−20.5781685539899606873798390906, −20.310792388648290034113977248718, −18.8625991826926750002487525902, −18.55893726354809305480430727517, −17.97250991508689935497748468064, −17.12819849447612660761923231154, −16.25748420472513630664078178915, −15.48992469465146342063560301314, −14.701490236577902316523481053540, −14.02496908931062833226139549164, −13.22321389609670654252070579214, −12.386483728703272921731212219205, −11.11744609611793191293387130647, −10.60935569955427799666368130343, −9.72683678000597867992328133611, −9.10619797662563396711170315552, −8.33987942601948515244255839597, −7.67675615206034244427228143353, −6.98044913662134307808826752458, −5.71618534151867721435796912163, −5.083472801504784682869872165352, −3.47239580229355811811926844915, −2.78163034232856308258207735199, −1.69554428286471351478712339322, −1.31182087030181335428045920902,
1.31182087030181335428045920902, 1.69554428286471351478712339322, 2.78163034232856308258207735199, 3.47239580229355811811926844915, 5.083472801504784682869872165352, 5.71618534151867721435796912163, 6.98044913662134307808826752458, 7.67675615206034244427228143353, 8.33987942601948515244255839597, 9.10619797662563396711170315552, 9.72683678000597867992328133611, 10.60935569955427799666368130343, 11.11744609611793191293387130647, 12.386483728703272921731212219205, 13.22321389609670654252070579214, 14.02496908931062833226139549164, 14.701490236577902316523481053540, 15.48992469465146342063560301314, 16.25748420472513630664078178915, 17.12819849447612660761923231154, 17.97250991508689935497748468064, 18.55893726354809305480430727517, 18.8625991826926750002487525902, 20.310792388648290034113977248718, 20.5781685539899606873798390906
