| L(s) = 1 | + 0.427·2-s − 2.84·3-s − 1.81·4-s − 0.938·5-s − 1.21·6-s + 1.20·7-s − 1.63·8-s + 5.07·9-s − 0.401·10-s − 1.74·11-s + 5.16·12-s + 13-s + 0.516·14-s + 2.66·15-s + 2.93·16-s + 3.52·17-s + 2.16·18-s + 7.44·19-s + 1.70·20-s − 3.43·21-s − 0.746·22-s − 2.87·23-s + 4.63·24-s − 4.11·25-s + 0.427·26-s − 5.88·27-s − 2.19·28-s + ⋯ |
| L(s) = 1 | + 0.302·2-s − 1.64·3-s − 0.908·4-s − 0.419·5-s − 0.495·6-s + 0.456·7-s − 0.576·8-s + 1.69·9-s − 0.126·10-s − 0.526·11-s + 1.49·12-s + 0.277·13-s + 0.138·14-s + 0.688·15-s + 0.734·16-s + 0.856·17-s + 0.510·18-s + 1.70·19-s + 0.381·20-s − 0.748·21-s − 0.159·22-s − 0.600·23-s + 0.946·24-s − 0.823·25-s + 0.0838·26-s − 1.13·27-s − 0.414·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1339 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
| good | 2 | \( 1 - 0.427T + 2T^{2} \) |
| 3 | \( 1 + 2.84T + 3T^{2} \) |
| 5 | \( 1 + 0.938T + 5T^{2} \) |
| 7 | \( 1 - 1.20T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 19 | \( 1 - 7.44T + 19T^{2} \) |
| 23 | \( 1 + 2.87T + 23T^{2} \) |
| 29 | \( 1 + 10.5T + 29T^{2} \) |
| 31 | \( 1 - 6.95T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 4.44T + 41T^{2} \) |
| 43 | \( 1 + 10.4T + 43T^{2} \) |
| 47 | \( 1 + 4.48T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 5.15T + 59T^{2} \) |
| 61 | \( 1 + 0.142T + 61T^{2} \) |
| 67 | \( 1 - 9.26T + 67T^{2} \) |
| 71 | \( 1 - 11.5T + 71T^{2} \) |
| 73 | \( 1 + 8.83T + 73T^{2} \) |
| 79 | \( 1 - 4.33T + 79T^{2} \) |
| 83 | \( 1 + 3.55T + 83T^{2} \) |
| 89 | \( 1 + 7.86T + 89T^{2} \) |
| 97 | \( 1 + 1.01T + 97T^{2} \) |
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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
−9.655935861331529186369095578097, −8.092211408748205075115674702769, −7.71659804243384035039958851628, −6.45210598470686508756622340303, −5.53204706548118492318081734730, −5.21175804320763648532844570958, −4.29761835746402958086411833684, −3.35482508507788206405462037560, −1.25548739073271915962991864772, 0,
1.25548739073271915962991864772, 3.35482508507788206405462037560, 4.29761835746402958086411833684, 5.21175804320763648532844570958, 5.53204706548118492318081734730, 6.45210598470686508756622340303, 7.71659804243384035039958851628, 8.092211408748205075115674702769, 9.655935861331529186369095578097
