| L(s) = 1 | + 1.73·2-s + 0.999·4-s − 5-s − 2.73·7-s − 1.73·8-s − 1.73·10-s − 4.73·11-s + 0.732·13-s − 4.73·14-s − 5·16-s + 19-s − 0.999·20-s − 8.19·22-s − 3.46·23-s + 25-s + 1.26·26-s − 2.73·28-s − 8.19·29-s + 8.92·31-s − 5.19·32-s + 2.73·35-s − 6.19·37-s + 1.73·38-s + 1.73·40-s − 1.26·41-s + 4.19·43-s − 4.73·44-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.499·4-s − 0.447·5-s − 1.03·7-s − 0.612·8-s − 0.547·10-s − 1.42·11-s + 0.203·13-s − 1.26·14-s − 1.25·16-s + 0.229·19-s − 0.223·20-s − 1.74·22-s − 0.722·23-s + 0.200·25-s + 0.248·26-s − 0.516·28-s − 1.52·29-s + 1.60·31-s − 0.918·32-s + 0.461·35-s − 1.01·37-s + 0.280·38-s + 0.273·40-s − 0.198·41-s + 0.639·43-s − 0.713·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 + 2.73T + 7T^{2} \) |
| 11 | \( 1 + 4.73T + 11T^{2} \) |
| 13 | \( 1 - 0.732T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 8.19T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 + 6.19T + 37T^{2} \) |
| 41 | \( 1 + 1.26T + 41T^{2} \) |
| 43 | \( 1 - 4.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 - 9.46T + 53T^{2} \) |
| 59 | \( 1 + 2.53T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 4.39T + 71T^{2} \) |
| 73 | \( 1 + 16.9T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 + 6.19T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.883189261924100285367699278865, −8.918811508542577170579471553856, −7.925284616352481666971539413070, −6.96919026304098461403641387292, −5.98955303734139928413255293136, −5.31448895197748011351559390899, −4.26098337197706662680269410074, −3.38757928029076444734609280301, −2.55116332079524321306653284472, 0,
2.55116332079524321306653284472, 3.38757928029076444734609280301, 4.26098337197706662680269410074, 5.31448895197748011351559390899, 5.98955303734139928413255293136, 6.96919026304098461403641387292, 7.925284616352481666971539413070, 8.918811508542577170579471553856, 9.883189261924100285367699278865
