| L(s) = 1 | + 21.4·2-s − 32.0·3-s − 52.9·4-s − 1.49e3·5-s − 687.·6-s − 1.22e3·7-s − 1.21e4·8-s − 1.86e4·9-s − 3.20e4·10-s + 5.99e4·11-s + 1.69e3·12-s + 1.43e5·13-s − 2.61e4·14-s + 4.79e4·15-s − 2.32e5·16-s − 3.99e5·18-s + 8.97e5·19-s + 7.90e4·20-s + 3.91e4·21-s + 1.28e6·22-s − 1.66e6·23-s + 3.88e5·24-s + 2.77e5·25-s + 3.06e6·26-s + 1.22e6·27-s + 6.46e4·28-s + 4.05e6·29-s + ⋯ |
| L(s) = 1 | + 0.946·2-s − 0.228·3-s − 0.103·4-s − 1.06·5-s − 0.216·6-s − 0.192·7-s − 1.04·8-s − 0.947·9-s − 1.01·10-s + 1.23·11-s + 0.0236·12-s + 1.38·13-s − 0.181·14-s + 0.244·15-s − 0.885·16-s − 0.897·18-s + 1.58·19-s + 0.110·20-s + 0.0439·21-s + 1.16·22-s − 1.23·23-s + 0.238·24-s + 0.142·25-s + 1.31·26-s + 0.445·27-s + 0.0198·28-s + 1.06·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 - 21.4T + 512T^{2} \) |
| 3 | \( 1 + 32.0T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.49e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.22e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.99e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.43e5T + 1.06e10T^{2} \) |
| 19 | \( 1 - 8.97e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.66e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.05e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.88e5T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.68e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 1.69e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 6.52e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.90e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 1.27e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 4.63e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.21e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.39e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 5.23e7T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.27e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 3.57e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 4.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.27e8T + 7.60e17T^{2} \) |
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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
−9.654519312324637433676645011532, −8.727030174537684755246218820648, −7.900786509845263145526456085584, −6.41693792817195811796925479224, −5.82920842495140974361356059608, −4.55854741668999026993024488548, −3.70967412881012797341998551068, −3.08332377638291785695157135810, −1.10252963139646886144569026978, 0,
1.10252963139646886144569026978, 3.08332377638291785695157135810, 3.70967412881012797341998551068, 4.55854741668999026993024488548, 5.82920842495140974361356059608, 6.41693792817195811796925479224, 7.900786509845263145526456085584, 8.727030174537684755246218820648, 9.654519312324637433676645011532
