| L(s) = 1 | + 2-s − 0.851·3-s + 4-s − 2.73·5-s − 0.851·6-s + 0.590·7-s + 8-s − 2.27·9-s − 2.73·10-s − 1.85·11-s − 0.851·12-s + 1.88·13-s + 0.590·14-s + 2.32·15-s + 16-s − 4.96·17-s − 2.27·18-s + 19-s − 2.73·20-s − 0.502·21-s − 1.85·22-s + 9.18·23-s − 0.851·24-s + 2.48·25-s + 1.88·26-s + 4.49·27-s + 0.590·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.491·3-s + 0.5·4-s − 1.22·5-s − 0.347·6-s + 0.223·7-s + 0.353·8-s − 0.758·9-s − 0.865·10-s − 0.558·11-s − 0.245·12-s + 0.522·13-s + 0.157·14-s + 0.601·15-s + 0.250·16-s − 1.20·17-s − 0.536·18-s + 0.229·19-s − 0.611·20-s − 0.109·21-s − 0.394·22-s + 1.91·23-s − 0.173·24-s + 0.497·25-s + 0.369·26-s + 0.864·27-s + 0.111·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 + T \) |
| good | 3 | \( 1 + 0.851T + 3T^{2} \) |
| 5 | \( 1 + 2.73T + 5T^{2} \) |
| 7 | \( 1 - 0.590T + 7T^{2} \) |
| 11 | \( 1 + 1.85T + 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 + 4.96T + 17T^{2} \) |
| 23 | \( 1 - 9.18T + 23T^{2} \) |
| 29 | \( 1 - 5.21T + 29T^{2} \) |
| 31 | \( 1 - 2.14T + 31T^{2} \) |
| 37 | \( 1 - 4.56T + 37T^{2} \) |
| 41 | \( 1 - 2.66T + 41T^{2} \) |
| 43 | \( 1 + 2.82T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 4.12T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 - 2.88T + 61T^{2} \) |
| 67 | \( 1 + 5.02T + 67T^{2} \) |
| 71 | \( 1 - 3.68T + 71T^{2} \) |
| 73 | \( 1 + 2.34T + 73T^{2} \) |
| 79 | \( 1 + 5.28T + 79T^{2} \) |
| 83 | \( 1 - 4.69T + 83T^{2} \) |
| 89 | \( 1 - 11.7T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−7.41765707191983221109135765313, −6.62837271070890845291246914099, −6.18492072252277956839984100198, −5.07358775723205460010241551408, −4.84777560510575433126314739149, −4.00123695821174925506221624418, −3.13574735629131488798070248610, −2.59210114245507183938696641081, −1.13775712449719920279470983473, 0,
1.13775712449719920279470983473, 2.59210114245507183938696641081, 3.13574735629131488798070248610, 4.00123695821174925506221624418, 4.84777560510575433126314739149, 5.07358775723205460010241551408, 6.18492072252277956839984100198, 6.62837271070890845291246914099, 7.41765707191983221109135765313
