| L(s) = 1 | + 1.32·2-s + 0.816·3-s − 0.244·4-s − 1.13·5-s + 1.08·6-s + 0.353·7-s − 2.97·8-s − 2.33·9-s − 1.49·10-s + 11-s − 0.199·12-s + 0.567·13-s + 0.467·14-s − 0.923·15-s − 3.45·16-s − 5.63·17-s − 3.09·18-s + 4.71·19-s + 0.276·20-s + 0.288·21-s + 1.32·22-s − 1.45·23-s − 2.42·24-s − 3.72·25-s + 0.752·26-s − 4.35·27-s − 0.0863·28-s + ⋯ |
| L(s) = 1 | + 0.936·2-s + 0.471·3-s − 0.122·4-s − 0.505·5-s + 0.441·6-s + 0.133·7-s − 1.05·8-s − 0.777·9-s − 0.473·10-s + 0.301·11-s − 0.0576·12-s + 0.157·13-s + 0.125·14-s − 0.238·15-s − 0.862·16-s − 1.36·17-s − 0.728·18-s + 1.08·19-s + 0.0618·20-s + 0.0629·21-s + 0.282·22-s − 0.302·23-s − 0.495·24-s − 0.744·25-s + 0.147·26-s − 0.838·27-s − 0.0163·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 2 | \( 1 - 1.32T + 2T^{2} \) |
| 3 | \( 1 - 0.816T + 3T^{2} \) |
| 5 | \( 1 + 1.13T + 5T^{2} \) |
| 7 | \( 1 - 0.353T + 7T^{2} \) |
| 13 | \( 1 - 0.567T + 13T^{2} \) |
| 17 | \( 1 + 5.63T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 + 1.45T + 23T^{2} \) |
| 29 | \( 1 + 3.70T + 29T^{2} \) |
| 31 | \( 1 + 6.22T + 31T^{2} \) |
| 37 | \( 1 + 5.83T + 37T^{2} \) |
| 41 | \( 1 + 1.85T + 41T^{2} \) |
| 43 | \( 1 - 3.51T + 43T^{2} \) |
| 47 | \( 1 - 3.12T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 4.57T + 59T^{2} \) |
| 61 | \( 1 - 2.35T + 61T^{2} \) |
| 67 | \( 1 - 11.9T + 67T^{2} \) |
| 71 | \( 1 - 8.13T + 71T^{2} \) |
| 73 | \( 1 - 1.48T + 73T^{2} \) |
| 79 | \( 1 - 7.57T + 79T^{2} \) |
| 83 | \( 1 + 9.80T + 83T^{2} \) |
| 89 | \( 1 - 7.94T + 89T^{2} \) |
| 97 | \( 1 + 10.9T + 97T^{2} \) |
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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
−9.094500530403423988014896528796, −8.379561090326052771491884979165, −7.55625715541569632451895502674, −6.49767238167705652895989224780, −5.63779156516819380525300768263, −4.85140725155699626151299916139, −3.83550782580066361344940696591, −3.32274694546197040649527481506, −2.10904493407157234446848688769, 0,
2.10904493407157234446848688769, 3.32274694546197040649527481506, 3.83550782580066361344940696591, 4.85140725155699626151299916139, 5.63779156516819380525300768263, 6.49767238167705652895989224780, 7.55625715541569632451895502674, 8.379561090326052771491884979165, 9.094500530403423988014896528796
