| L(s) = 1 | − 16.5·2-s + 147.·4-s − 114.·5-s + 1.43e3·7-s − 320.·8-s + 1.89e3·10-s − 5.92e3·11-s − 1.14e4·13-s − 2.38e4·14-s − 1.35e4·16-s + 2.02e4·17-s − 6.35e3·19-s − 1.68e4·20-s + 9.83e4·22-s − 7.58e4·23-s − 6.50e4·25-s + 1.89e5·26-s + 2.11e5·28-s + 7.47e4·29-s − 1.89e5·31-s + 2.65e5·32-s − 3.35e5·34-s − 1.64e5·35-s − 3.34e4·37-s + 1.05e5·38-s + 3.66e4·40-s − 1.41e5·41-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.15·4-s − 0.408·5-s + 1.58·7-s − 0.221·8-s + 0.599·10-s − 1.34·11-s − 1.44·13-s − 2.32·14-s − 0.826·16-s + 0.998·17-s − 0.212·19-s − 0.470·20-s + 1.96·22-s − 1.29·23-s − 0.833·25-s + 2.11·26-s + 1.82·28-s + 0.569·29-s − 1.14·31-s + 1.43·32-s − 1.46·34-s − 0.647·35-s − 0.108·37-s + 0.311·38-s + 0.0905·40-s − 0.320·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 16.5T + 128T^{2} \) |
| 5 | \( 1 + 114.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.92e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.14e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.02e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 6.35e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 7.47e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.89e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 3.34e4T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.41e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 2.46e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.35e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.65e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 2.04e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 5.90e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.35e4T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.95e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 8.17e5T + 1.10e13T^{2} \) |
| 79 | \( 1 - 7.57e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.01e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 1.37e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.06e7T + 8.07e13T^{2} \) |
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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
−15.44300699008772398870322230485, −14.16771241367756159284030634661, −12.12825516654550966983914809766, −10.89419681588924115655180323773, −9.823570931174348196955616277347, −8.035162359291653759836583903955, −7.69797392899539787341016412379, −4.96483972803010063458016663411, −1.96325339036313430748040369469, 0,
1.96325339036313430748040369469, 4.96483972803010063458016663411, 7.69797392899539787341016412379, 8.035162359291653759836583903955, 9.823570931174348196955616277347, 10.89419681588924115655180323773, 12.12825516654550966983914809766, 14.16771241367756159284030634661, 15.44300699008772398870322230485
