| L(s) = 1 | − 20.8·2-s − 27·3-s + 305.·4-s + 146.·5-s + 562.·6-s + 343·7-s − 3.69e3·8-s + 729·9-s − 3.04e3·10-s − 7.49e3·11-s − 8.24e3·12-s + 8.55e3·13-s − 7.14e3·14-s − 3.95e3·15-s + 3.77e4·16-s − 5.21e3·17-s − 1.51e4·18-s − 4.71e4·19-s + 4.46e4·20-s − 9.26e3·21-s + 1.55e5·22-s + 2.38e4·23-s + 9.96e4·24-s − 5.67e4·25-s − 1.78e5·26-s − 1.96e4·27-s + 1.04e5·28-s + ⋯ |
| L(s) = 1 | − 1.83·2-s − 0.577·3-s + 2.38·4-s + 0.523·5-s + 1.06·6-s + 0.377·7-s − 2.54·8-s + 0.333·9-s − 0.963·10-s − 1.69·11-s − 1.37·12-s + 1.07·13-s − 0.695·14-s − 0.302·15-s + 2.30·16-s − 0.257·17-s − 0.613·18-s − 1.57·19-s + 1.24·20-s − 0.218·21-s + 3.12·22-s + 0.408·23-s + 1.47·24-s − 0.725·25-s − 1.98·26-s − 0.192·27-s + 0.901·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 21 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 27T \) |
| 7 | \( 1 - 343T \) |
| good | 2 | \( 1 + 20.8T + 128T^{2} \) |
| 5 | \( 1 - 146.T + 7.81e4T^{2} \) |
| 11 | \( 1 + 7.49e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 8.55e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 5.21e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 4.71e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.83e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 1.80e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.04e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 6.32e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.43e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 5.24e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 5.20e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.42e6T + 2.48e12T^{2} \) |
| 61 | \( 1 + 6.16e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.63e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 5.95e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.41e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 1.08e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 8.80e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 6.13e6T + 8.07e13T^{2} \) |
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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
−16.35376414830336104329946876990, −15.33906116366350644198804511681, −12.98702501814532961441949753557, −11.06864907834849141420315822462, −10.48357678196108462687760349283, −8.923696557726720156659706900536, −7.59432258364094413923615516970, −5.91638510993332172401398668443, −1.94579027207014780314016043593, 0,
1.94579027207014780314016043593, 5.91638510993332172401398668443, 7.59432258364094413923615516970, 8.923696557726720156659706900536, 10.48357678196108462687760349283, 11.06864907834849141420315822462, 12.98702501814532961441949753557, 15.33906116366350644198804511681, 16.35376414830336104329946876990
