| L(s) = 1 | − 2.18·2-s + 2.75·4-s − 2.81·5-s + 0.316·7-s − 1.65·8-s + 6.13·10-s + 1.81·11-s + 0.628·13-s − 0.689·14-s − 1.90·16-s − 5.16·17-s + 4.02·19-s − 7.75·20-s − 3.96·22-s + 0.344·23-s + 2.90·25-s − 1.37·26-s + 0.871·28-s − 4.49·29-s + 3.75·31-s + 7.46·32-s + 11.2·34-s − 0.888·35-s + 5.12·37-s − 8.77·38-s + 4.65·40-s + 3.71·41-s + ⋯ |
| L(s) = 1 | − 1.54·2-s + 1.37·4-s − 1.25·5-s + 0.119·7-s − 0.585·8-s + 1.93·10-s + 0.547·11-s + 0.174·13-s − 0.184·14-s − 0.476·16-s − 1.25·17-s + 0.923·19-s − 1.73·20-s − 0.844·22-s + 0.0718·23-s + 0.580·25-s − 0.268·26-s + 0.164·28-s − 0.834·29-s + 0.674·31-s + 1.32·32-s + 1.93·34-s − 0.150·35-s + 0.841·37-s − 1.42·38-s + 0.735·40-s + 0.579·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 2 | \( 1 + 2.18T + 2T^{2} \) |
| 5 | \( 1 + 2.81T + 5T^{2} \) |
| 7 | \( 1 - 0.316T + 7T^{2} \) |
| 11 | \( 1 - 1.81T + 11T^{2} \) |
| 13 | \( 1 - 0.628T + 13T^{2} \) |
| 17 | \( 1 + 5.16T + 17T^{2} \) |
| 19 | \( 1 - 4.02T + 19T^{2} \) |
| 23 | \( 1 - 0.344T + 23T^{2} \) |
| 29 | \( 1 + 4.49T + 29T^{2} \) |
| 31 | \( 1 - 3.75T + 31T^{2} \) |
| 37 | \( 1 - 5.12T + 37T^{2} \) |
| 41 | \( 1 - 3.71T + 41T^{2} \) |
| 43 | \( 1 + 5.00T + 43T^{2} \) |
| 47 | \( 1 + 5.40T + 47T^{2} \) |
| 53 | \( 1 + 9.46T + 53T^{2} \) |
| 59 | \( 1 - 3.11T + 59T^{2} \) |
| 61 | \( 1 - 6.73T + 61T^{2} \) |
| 67 | \( 1 - 0.908T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 13.1T + 73T^{2} \) |
| 79 | \( 1 + 3.13T + 79T^{2} \) |
| 83 | \( 1 + 8.80T + 83T^{2} \) |
| 89 | \( 1 - 4.68T + 89T^{2} \) |
| 97 | \( 1 + 2.28T + 97T^{2} \) |
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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
−8.142919228139622836423835928474, −7.63369943040827016917834886709, −6.90530559553455455544201350332, −6.31199210099859439089042234343, −4.95837435515822449842641451050, −4.18870845325579870251499124277, −3.31547895650486363483461144057, −2.13911788108161911283994003737, −1.05035696078502528477409049683, 0,
1.05035696078502528477409049683, 2.13911788108161911283994003737, 3.31547895650486363483461144057, 4.18870845325579870251499124277, 4.95837435515822449842641451050, 6.31199210099859439089042234343, 6.90530559553455455544201350332, 7.63369943040827016917834886709, 8.142919228139622836423835928474
