| L(s) = 1 | + 0.164·2-s − 31.9·4-s + 37.9·5-s − 43.8·7-s − 10.5·8-s + 6.24·10-s + 163.·11-s + 430.·13-s − 7.21·14-s + 1.02e3·16-s − 740.·17-s − 916.·19-s − 1.21e3·20-s + 26.8·22-s + 529·23-s − 1.68e3·25-s + 70.7·26-s + 1.40e3·28-s + 5.11e3·29-s − 5.70e3·31-s + 504.·32-s − 121.·34-s − 1.66e3·35-s − 1.09e4·37-s − 150.·38-s − 399.·40-s − 1.10e4·41-s + ⋯ |
| L(s) = 1 | + 0.0290·2-s − 0.999·4-s + 0.679·5-s − 0.338·7-s − 0.0581·8-s + 0.0197·10-s + 0.406·11-s + 0.706·13-s − 0.00983·14-s + 0.997·16-s − 0.621·17-s − 0.582·19-s − 0.678·20-s + 0.0118·22-s + 0.208·23-s − 0.538·25-s + 0.0205·26-s + 0.337·28-s + 1.12·29-s − 1.06·31-s + 0.0871·32-s − 0.0180·34-s − 0.229·35-s − 1.31·37-s − 0.0169·38-s − 0.0395·40-s − 1.03·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 207 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - 529T \) |
| good | 2 | \( 1 - 0.164T + 32T^{2} \) |
| 5 | \( 1 - 37.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 43.8T + 1.68e4T^{2} \) |
| 11 | \( 1 - 163.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 430.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 740.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 916.T + 2.47e6T^{2} \) |
| 29 | \( 1 - 5.11e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.70e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.09e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.10e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.52e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.44e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 6.85e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.15e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.19e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.51e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.40e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.33e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.42e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.14e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 6.34e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.19e4T + 8.58e9T^{2} \) |
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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
−10.86948888896119107689644349951, −9.911929381559681482478992681551, −9.069745384841626583561847170253, −8.307776231421211523724371681262, −6.72880009208883527409861210342, −5.74892149779820994377155910429, −4.56073626258950851500444745583, −3.38122048581652377898378306264, −1.60799706400692748509275620964, 0,
1.60799706400692748509275620964, 3.38122048581652377898378306264, 4.56073626258950851500444745583, 5.74892149779820994377155910429, 6.72880009208883527409861210342, 8.307776231421211523724371681262, 9.069745384841626583561847170253, 9.911929381559681482478992681551, 10.86948888896119107689644349951
