| L(s) = 1 | − 0.899·2-s + 0.683·3-s − 1.19·4-s − 2.15·5-s − 0.615·6-s + 0.912·7-s + 2.87·8-s − 2.53·9-s + 1.93·10-s + 11-s − 0.813·12-s + 4.57·13-s − 0.820·14-s − 1.47·15-s − 0.202·16-s + 4.89·17-s + 2.27·18-s − 5.49·19-s + 2.56·20-s + 0.623·21-s − 0.899·22-s − 5.93·23-s + 1.96·24-s − 0.354·25-s − 4.11·26-s − 3.78·27-s − 1.08·28-s + ⋯ |
| L(s) = 1 | − 0.636·2-s + 0.394·3-s − 0.595·4-s − 0.963·5-s − 0.251·6-s + 0.344·7-s + 1.01·8-s − 0.844·9-s + 0.613·10-s + 0.301·11-s − 0.234·12-s + 1.26·13-s − 0.219·14-s − 0.380·15-s − 0.0506·16-s + 1.18·17-s + 0.537·18-s − 1.25·19-s + 0.573·20-s + 0.136·21-s − 0.191·22-s − 1.23·23-s + 0.400·24-s − 0.0709·25-s − 0.806·26-s − 0.728·27-s − 0.205·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 869 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 + 0.899T + 2T^{2} \) |
| 3 | \( 1 - 0.683T + 3T^{2} \) |
| 5 | \( 1 + 2.15T + 5T^{2} \) |
| 7 | \( 1 - 0.912T + 7T^{2} \) |
| 13 | \( 1 - 4.57T + 13T^{2} \) |
| 17 | \( 1 - 4.89T + 17T^{2} \) |
| 19 | \( 1 + 5.49T + 19T^{2} \) |
| 23 | \( 1 + 5.93T + 23T^{2} \) |
| 29 | \( 1 - 2.56T + 29T^{2} \) |
| 31 | \( 1 + 5.48T + 31T^{2} \) |
| 37 | \( 1 + 5.67T + 37T^{2} \) |
| 41 | \( 1 + 0.630T + 41T^{2} \) |
| 43 | \( 1 + 3.70T + 43T^{2} \) |
| 47 | \( 1 + 10.8T + 47T^{2} \) |
| 53 | \( 1 + 4.26T + 53T^{2} \) |
| 59 | \( 1 - 10.0T + 59T^{2} \) |
| 61 | \( 1 - 7.18T + 61T^{2} \) |
| 67 | \( 1 + 11.9T + 67T^{2} \) |
| 71 | \( 1 - 5.12T + 71T^{2} \) |
| 73 | \( 1 + 8.12T + 73T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 6.94T + 89T^{2} \) |
| 97 | \( 1 - 1.89T + 97T^{2} \) |
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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
−9.605487093832011723664617804211, −8.506767207431864523251627209758, −8.368815546679778203161032940482, −7.65305172449621086314792620742, −6.34136977603629435124744716736, −5.25530639904935818632145501982, −4.03148523821402044686470102057, −3.47485495193469566340502224738, −1.65013840766599444194991308911, 0,
1.65013840766599444194991308911, 3.47485495193469566340502224738, 4.03148523821402044686470102057, 5.25530639904935818632145501982, 6.34136977603629435124744716736, 7.65305172449621086314792620742, 8.368815546679778203161032940482, 8.506767207431864523251627209758, 9.605487093832011723664617804211
