| L(s) = 1 | + 2-s + 4-s − 5-s − 4.62·7-s + 8-s − 10-s + 4.10·11-s + 0.710·13-s − 4.62·14-s + 16-s + 3.20·17-s + 0.710·19-s − 20-s + 4.10·22-s − 8.54·23-s + 25-s + 0.710·26-s − 4.62·28-s + 7.91·29-s − 2.44·31-s + 32-s + 3.20·34-s + 4.62·35-s − 37-s + 0.710·38-s − 40-s + 7.86·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.74·7-s + 0.353·8-s − 0.316·10-s + 1.23·11-s + 0.197·13-s − 1.23·14-s + 0.250·16-s + 0.777·17-s + 0.163·19-s − 0.223·20-s + 0.874·22-s − 1.78·23-s + 0.200·25-s + 0.139·26-s − 0.874·28-s + 1.47·29-s − 0.438·31-s + 0.176·32-s + 0.549·34-s + 0.782·35-s − 0.164·37-s + 0.115·38-s − 0.158·40-s + 1.22·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.298097175\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.298097175\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 4.62T + 7T^{2} \) |
| 11 | \( 1 - 4.10T + 11T^{2} \) |
| 13 | \( 1 - 0.710T + 13T^{2} \) |
| 17 | \( 1 - 3.20T + 17T^{2} \) |
| 19 | \( 1 - 0.710T + 19T^{2} \) |
| 23 | \( 1 + 8.54T + 23T^{2} \) |
| 29 | \( 1 - 7.91T + 29T^{2} \) |
| 31 | \( 1 + 2.44T + 31T^{2} \) |
| 41 | \( 1 - 7.86T + 41T^{2} \) |
| 43 | \( 1 - 2.60T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 - 9.15T + 53T^{2} \) |
| 59 | \( 1 - 0.951T + 59T^{2} \) |
| 61 | \( 1 + 3.39T + 61T^{2} \) |
| 67 | \( 1 - 2.95T + 67T^{2} \) |
| 71 | \( 1 - 5.04T + 71T^{2} \) |
| 73 | \( 1 - 16.2T + 73T^{2} \) |
| 79 | \( 1 - 16.2T + 79T^{2} \) |
| 83 | \( 1 + 9.75T + 83T^{2} \) |
| 89 | \( 1 - 2.13T + 89T^{2} \) |
| 97 | \( 1 - 9.86T + 97T^{2} \) |
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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
−8.611885511462766392462437052994, −7.74138961444963928999422567854, −6.88583131551328938441836809870, −6.30033301619035904332769654085, −5.84058599965658617352732801210, −4.62687534487359993510672443936, −3.66558351564406813982063191157, −3.46863694327578630058221255007, −2.29121100399843943369615663000, −0.808697552858476107938226420465,
0.808697552858476107938226420465, 2.29121100399843943369615663000, 3.46863694327578630058221255007, 3.66558351564406813982063191157, 4.62687534487359993510672443936, 5.84058599965658617352732801210, 6.30033301619035904332769654085, 6.88583131551328938441836809870, 7.74138961444963928999422567854, 8.611885511462766392462437052994
