| L(s) = 1 | − 1.07·2-s + 5.02·3-s − 6.85·4-s − 4.32·5-s − 5.37·6-s − 4.44·7-s + 15.9·8-s − 1.79·9-s + 4.63·10-s + 68.9·11-s − 34.4·12-s − 27.1·13-s + 4.76·14-s − 21.7·15-s + 37.7·16-s + 1.92·18-s − 123.·19-s + 29.6·20-s − 22.3·21-s − 73.8·22-s − 98.6·23-s + 79.8·24-s − 106.·25-s + 29.1·26-s − 144.·27-s + 30.4·28-s − 21.9·29-s + ⋯ |
| L(s) = 1 | − 0.378·2-s + 0.966·3-s − 0.856·4-s − 0.386·5-s − 0.365·6-s − 0.240·7-s + 0.703·8-s − 0.0666·9-s + 0.146·10-s + 1.89·11-s − 0.827·12-s − 0.580·13-s + 0.0909·14-s − 0.373·15-s + 0.590·16-s + 0.0252·18-s − 1.49·19-s + 0.331·20-s − 0.231·21-s − 0.715·22-s − 0.894·23-s + 0.679·24-s − 0.850·25-s + 0.219·26-s − 1.03·27-s + 0.205·28-s − 0.140·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 1.07T + 8T^{2} \) |
| 3 | \( 1 - 5.02T + 27T^{2} \) |
| 5 | \( 1 + 4.32T + 125T^{2} \) |
| 7 | \( 1 + 4.44T + 343T^{2} \) |
| 11 | \( 1 - 68.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 27.1T + 2.19e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 98.6T + 1.21e4T^{2} \) |
| 29 | \( 1 + 21.9T + 2.43e4T^{2} \) |
| 31 | \( 1 + 241.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 324.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 164.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 383.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 411.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 63.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 37.1T + 2.26e5T^{2} \) |
| 67 | \( 1 + 48.2T + 3.00e5T^{2} \) |
| 71 | \( 1 + 672.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 562.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 461.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 972.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 16.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 598.T + 9.12e5T^{2} \) |
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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
−10.72397954041437808390348829424, −9.485564241336014803089092068659, −9.071333279892593275986575817198, −8.257089469587002254314172203679, −7.29242022310052385078250297462, −5.94828856336420709573710012188, −4.24956837226630134131245431077, −3.65879524289308241778556071881, −1.87972748143915294414097184787, 0,
1.87972748143915294414097184787, 3.65879524289308241778556071881, 4.24956837226630134131245431077, 5.94828856336420709573710012188, 7.29242022310052385078250297462, 8.257089469587002254314172203679, 9.071333279892593275986575817198, 9.485564241336014803089092068659, 10.72397954041437808390348829424
