| L(s) = 1 | + 1.09·2-s + 1.45·3-s − 0.809·4-s + 1.35·5-s + 1.59·6-s + 0.216·7-s − 3.06·8-s − 0.869·9-s + 1.47·10-s − 6.03·11-s − 1.18·12-s − 2.87·13-s + 0.236·14-s + 1.97·15-s − 1.72·16-s + 0.282·17-s − 0.948·18-s − 1.13·19-s − 1.09·20-s + 0.315·21-s − 6.57·22-s − 3.94·23-s − 4.47·24-s − 3.17·25-s − 3.13·26-s − 5.64·27-s − 0.175·28-s + ⋯ |
| L(s) = 1 | + 0.771·2-s + 0.842·3-s − 0.404·4-s + 0.604·5-s + 0.650·6-s + 0.0817·7-s − 1.08·8-s − 0.289·9-s + 0.466·10-s − 1.81·11-s − 0.341·12-s − 0.797·13-s + 0.0630·14-s + 0.509·15-s − 0.431·16-s + 0.0685·17-s − 0.223·18-s − 0.260·19-s − 0.244·20-s + 0.0688·21-s − 1.40·22-s − 0.822·23-s − 0.913·24-s − 0.635·25-s − 0.615·26-s − 1.08·27-s − 0.0331·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1849 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 43 | \( 1 \) |
| good | 2 | \( 1 - 1.09T + 2T^{2} \) |
| 3 | \( 1 - 1.45T + 3T^{2} \) |
| 5 | \( 1 - 1.35T + 5T^{2} \) |
| 7 | \( 1 - 0.216T + 7T^{2} \) |
| 11 | \( 1 + 6.03T + 11T^{2} \) |
| 13 | \( 1 + 2.87T + 13T^{2} \) |
| 17 | \( 1 - 0.282T + 17T^{2} \) |
| 19 | \( 1 + 1.13T + 19T^{2} \) |
| 23 | \( 1 + 3.94T + 23T^{2} \) |
| 29 | \( 1 - 6.90T + 29T^{2} \) |
| 31 | \( 1 + 9.89T + 31T^{2} \) |
| 37 | \( 1 - 7.54T + 37T^{2} \) |
| 41 | \( 1 - 5.12T + 41T^{2} \) |
| 47 | \( 1 - 0.462T + 47T^{2} \) |
| 53 | \( 1 - 8.35T + 53T^{2} \) |
| 59 | \( 1 + 8.35T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 + 6.29T + 67T^{2} \) |
| 71 | \( 1 + 1.47T + 71T^{2} \) |
| 73 | \( 1 + 0.820T + 73T^{2} \) |
| 79 | \( 1 - 6.15T + 79T^{2} \) |
| 83 | \( 1 + 7.20T + 83T^{2} \) |
| 89 | \( 1 - 4.43T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−8.822680001780424780945961743930, −8.075168826915989708882279290142, −7.49898990219666441290371238944, −6.10080740592762264606786682232, −5.51157266985080185827157228167, −4.79980155919322527314906245500, −3.79401045180829913669603729424, −2.74816715669244858187833209473, −2.27577841578509552289254211080, 0,
2.27577841578509552289254211080, 2.74816715669244858187833209473, 3.79401045180829913669603729424, 4.79980155919322527314906245500, 5.51157266985080185827157228167, 6.10080740592762264606786682232, 7.49898990219666441290371238944, 8.075168826915989708882279290142, 8.822680001780424780945961743930
