| L(s) = 1 | + 1.73·2-s + 3-s + 0.999·4-s + 5-s + 1.73·6-s + 0.732·7-s − 1.73·8-s + 9-s + 1.73·10-s + 1.26·11-s + 0.999·12-s − 2.73·13-s + 1.26·14-s + 15-s − 5·16-s + 1.73·18-s + 19-s + 0.999·20-s + 0.732·21-s + 2.19·22-s − 3.46·23-s − 1.73·24-s + 25-s − 4.73·26-s + 27-s + 0.732·28-s − 2.19·29-s + ⋯ |
| L(s) = 1 | + 1.22·2-s + 0.577·3-s + 0.499·4-s + 0.447·5-s + 0.707·6-s + 0.276·7-s − 0.612·8-s + 0.333·9-s + 0.547·10-s + 0.382·11-s + 0.288·12-s − 0.757·13-s + 0.338·14-s + 0.258·15-s − 1.25·16-s + 0.408·18-s + 0.229·19-s + 0.223·20-s + 0.159·21-s + 0.468·22-s − 0.722·23-s − 0.353·24-s + 0.200·25-s − 0.928·26-s + 0.192·27-s + 0.138·28-s − 0.407·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.668697205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.668697205\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 7 | \( 1 - 0.732T + 7T^{2} \) |
| 11 | \( 1 - 1.26T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 + 2.19T + 29T^{2} \) |
| 31 | \( 1 + 4.92T + 31T^{2} \) |
| 37 | \( 1 - 4.19T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 6.19T + 43T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 2.53T + 53T^{2} \) |
| 59 | \( 1 - 9.46T + 59T^{2} \) |
| 61 | \( 1 + 13.4T + 61T^{2} \) |
| 67 | \( 1 - 8T + 67T^{2} \) |
| 71 | \( 1 - 16.3T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 - 0.928T + 83T^{2} \) |
| 89 | \( 1 - 7.26T + 89T^{2} \) |
| 97 | \( 1 - 4.19T + 97T^{2} \) |
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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
−12.15146469924703303495839368295, −11.16406106427020001019077922362, −9.812863144240357561180810209366, −9.102257630028187947415108993731, −7.87435648646176746695740198796, −6.67728330568751329913910617748, −5.58359033619848602859785877093, −4.58467533642185988339614222074, −3.49393018214745799274320542749, −2.19944547378500630649591783378,
2.19944547378500630649591783378, 3.49393018214745799274320542749, 4.58467533642185988339614222074, 5.58359033619848602859785877093, 6.67728330568751329913910617748, 7.87435648646176746695740198796, 9.102257630028187947415108993731, 9.812863144240357561180810209366, 11.16406106427020001019077922362, 12.15146469924703303495839368295
