| L(s) = 1 | − 2.27·3-s − 5-s − 2.37·7-s + 2.16·9-s − 1.84·11-s + 0.00890·13-s + 2.27·15-s + 1.21·19-s + 5.39·21-s + 2.28·23-s + 25-s + 1.89·27-s − 9.19·29-s + 3.15·31-s + 4.20·33-s + 2.37·35-s + 3.17·37-s − 0.0202·39-s − 4.96·41-s − 3.34·43-s − 2.16·45-s + 8.96·47-s − 1.36·49-s + 12.5·53-s + 1.84·55-s − 2.75·57-s + 7.27·59-s + ⋯ |
| L(s) = 1 | − 1.31·3-s − 0.447·5-s − 0.897·7-s + 0.722·9-s − 0.557·11-s + 0.00247·13-s + 0.586·15-s + 0.277·19-s + 1.17·21-s + 0.475·23-s + 0.200·25-s + 0.364·27-s − 1.70·29-s + 0.567·31-s + 0.731·33-s + 0.401·35-s + 0.522·37-s − 0.00324·39-s − 0.775·41-s − 0.509·43-s − 0.323·45-s + 1.30·47-s − 0.195·49-s + 1.72·53-s + 0.249·55-s − 0.364·57-s + 0.946·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5780 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2.27T + 3T^{2} \) |
| 7 | \( 1 + 2.37T + 7T^{2} \) |
| 11 | \( 1 + 1.84T + 11T^{2} \) |
| 13 | \( 1 - 0.00890T + 13T^{2} \) |
| 19 | \( 1 - 1.21T + 19T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 + 9.19T + 29T^{2} \) |
| 31 | \( 1 - 3.15T + 31T^{2} \) |
| 37 | \( 1 - 3.17T + 37T^{2} \) |
| 41 | \( 1 + 4.96T + 41T^{2} \) |
| 43 | \( 1 + 3.34T + 43T^{2} \) |
| 47 | \( 1 - 8.96T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 7.27T + 59T^{2} \) |
| 61 | \( 1 - 7.98T + 61T^{2} \) |
| 67 | \( 1 - 2.82T + 67T^{2} \) |
| 71 | \( 1 - 2.02T + 71T^{2} \) |
| 73 | \( 1 + 12.5T + 73T^{2} \) |
| 79 | \( 1 - 13.5T + 79T^{2} \) |
| 83 | \( 1 - 1.05T + 83T^{2} \) |
| 89 | \( 1 + 1.15T + 89T^{2} \) |
| 97 | \( 1 + 15.7T + 97T^{2} \) |
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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
−7.51860899102024688372130126514, −6.96816088629940263073982997870, −6.29436602106093823756384770095, −5.55682607446599811861033360949, −5.10735624419936718079577873896, −4.12062639996893339590466937753, −3.35953477693229645378957334743, −2.37390606475612095074223001424, −0.906924513488275678733126590210, 0,
0.906924513488275678733126590210, 2.37390606475612095074223001424, 3.35953477693229645378957334743, 4.12062639996893339590466937753, 5.10735624419936718079577873896, 5.55682607446599811861033360949, 6.29436602106093823756384770095, 6.96816088629940263073982997870, 7.51860899102024688372130126514
