| L(s) = 1 | + 4·2-s − 20.4·3-s + 16·4-s + 34.4·5-s − 81.9·6-s − 18.9·7-s + 64·8-s + 176.·9-s + 137.·10-s + 349.·11-s − 327.·12-s − 711.·13-s − 75.6·14-s − 705.·15-s + 256·16-s + 221.·17-s + 705.·18-s + 551.·20-s + 387.·21-s + 1.39e3·22-s − 662.·23-s − 1.31e3·24-s − 1.93e3·25-s − 2.84e3·26-s + 1.36e3·27-s − 302.·28-s + 7.21e3·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.31·3-s + 0.5·4-s + 0.616·5-s − 0.929·6-s − 0.145·7-s + 0.353·8-s + 0.726·9-s + 0.435·10-s + 0.870·11-s − 0.656·12-s − 1.16·13-s − 0.103·14-s − 0.809·15-s + 0.250·16-s + 0.186·17-s + 0.513·18-s + 0.308·20-s + 0.191·21-s + 0.615·22-s − 0.261·23-s − 0.464·24-s − 0.620·25-s − 0.825·26-s + 0.359·27-s − 0.0729·28-s + 1.59·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 20.4T + 243T^{2} \) |
| 5 | \( 1 - 34.4T + 3.12e3T^{2} \) |
| 7 | \( 1 + 18.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 349.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 711.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 221.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 662.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.21e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.40e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.97e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.11e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 318.T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.72e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.11e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.79e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.74e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.49e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.17e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.47e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.61e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.17e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.45e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.95e4T + 8.58e9T^{2} \) |
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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
−9.639697000395356983664672259105, −8.232176697495258518185271667459, −6.90540272108608536310947990030, −6.47473839113900235586880530775, −5.51201723960271204536313721975, −4.95770561863875806426837089839, −3.87032186674611192676191852712, −2.49770640677928188718258621651, −1.29796450615623068183450847152, 0,
1.29796450615623068183450847152, 2.49770640677928188718258621651, 3.87032186674611192676191852712, 4.95770561863875806426837089839, 5.51201723960271204536313721975, 6.47473839113900235586880530775, 6.90540272108608536310947990030, 8.232176697495258518185271667459, 9.639697000395356983664672259105
