| L(s) = 1 | + 0.539·2-s − 3-s − 1.70·4-s + 0.170·5-s − 0.539·6-s + 7-s − 2·8-s + 9-s + 0.0917·10-s − 3.41·11-s + 1.70·12-s + 5.70·13-s + 0.539·14-s − 0.170·15-s + 2.34·16-s + 17-s + 0.539·18-s − 2.70·19-s − 0.290·20-s − 21-s − 1.84·22-s − 2.80·23-s + 2·24-s − 4.97·25-s + 3.07·26-s − 27-s − 1.70·28-s + ⋯ |
| L(s) = 1 | + 0.381·2-s − 0.577·3-s − 0.854·4-s + 0.0760·5-s − 0.220·6-s + 0.377·7-s − 0.707·8-s + 0.333·9-s + 0.0290·10-s − 1.03·11-s + 0.493·12-s + 1.58·13-s + 0.144·14-s − 0.0439·15-s + 0.585·16-s + 0.242·17-s + 0.127·18-s − 0.621·19-s − 0.0650·20-s − 0.218·21-s − 0.392·22-s − 0.584·23-s + 0.408·24-s − 0.994·25-s + 0.603·26-s − 0.192·27-s − 0.323·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| good | 2 | \( 1 - 0.539T + 2T^{2} \) |
| 5 | \( 1 - 0.170T + 5T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 + 3.41T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 19 | \( 1 + 2.70T + 19T^{2} \) |
| 23 | \( 1 + 2.80T + 23T^{2} \) |
| 29 | \( 1 + 3.41T + 29T^{2} \) |
| 31 | \( 1 + 0.581T + 31T^{2} \) |
| 37 | \( 1 + 2.65T + 37T^{2} \) |
| 41 | \( 1 - 8.83T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 3.53T + 47T^{2} \) |
| 53 | \( 1 - 12.6T + 53T^{2} \) |
| 59 | \( 1 - 3.87T + 59T^{2} \) |
| 61 | \( 1 + 8.26T + 61T^{2} \) |
| 67 | \( 1 + 12.2T + 67T^{2} \) |
| 71 | \( 1 + 6.87T + 71T^{2} \) |
| 73 | \( 1 + 3.90T + 73T^{2} \) |
| 83 | \( 1 - 13.2T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 + 4.24T + 97T^{2} \) |
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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
−8.030875713351296531657100111384, −7.50740653659740940106176097882, −6.17582799543902873977070063785, −5.82428178187939199254432164941, −5.17502540234050022118618138543, −4.18684094727970303192967189956, −3.79287620836382493931639511613, −2.53343308988690201442766771058, −1.26066758937245580898068466230, 0,
1.26066758937245580898068466230, 2.53343308988690201442766771058, 3.79287620836382493931639511613, 4.18684094727970303192967189956, 5.17502540234050022118618138543, 5.82428178187939199254432164941, 6.17582799543902873977070063785, 7.50740653659740940106176097882, 8.030875713351296531657100111384
