| L(s) = 1 | + 16.2·2-s + 134.·4-s + 312.·5-s + 107.·8-s + 5.05e3·10-s + 283.·11-s + 1.17e4·13-s − 1.54e4·16-s + 4.02e3·17-s − 3.78e3·19-s + 4.20e4·20-s + 4.59e3·22-s + 2.58e3·23-s + 1.92e4·25-s + 1.89e5·26-s + 1.27e5·29-s + 1.59e5·31-s − 2.64e5·32-s + 6.51e4·34-s + 5.83e5·37-s − 6.13e4·38-s + 3.34e4·40-s + 1.85e4·41-s − 3.23e5·43-s + 3.82e4·44-s + 4.18e4·46-s − 1.92e3·47-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.05·4-s + 1.11·5-s + 0.0740·8-s + 1.59·10-s + 0.0642·11-s + 1.47·13-s − 0.945·16-s + 0.198·17-s − 0.126·19-s + 1.17·20-s + 0.0920·22-s + 0.0442·23-s + 0.246·25-s + 2.11·26-s + 0.971·29-s + 0.958·31-s − 1.42·32-s + 0.284·34-s + 1.89·37-s − 0.181·38-s + 0.0826·40-s + 0.0420·41-s − 0.619·43-s + 0.0676·44-s + 0.0633·46-s − 0.00270·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.079875724\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.079875724\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 16.2T + 128T^{2} \) |
| 5 | \( 1 - 312.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 283.T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.17e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.02e3T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.78e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 2.58e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.27e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 1.59e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.83e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.85e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.23e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 1.92e3T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.35e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 8.68e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.27e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 5.84e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 5.67e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.97e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 8.70e5T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.87e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 1.02e7T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.98e6T + 8.07e13T^{2} \) |
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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
−10.05687926948563408464548325488, −9.138198028287935682656494669045, −8.111512982996246229039532821014, −6.49127282492691650653935217420, −6.16617490779982099980480234208, −5.22893474695512436346531564500, −4.25182304240399337595134213459, −3.22104381336452817800724765724, −2.22706506323772413245994434440, −1.01681578672332627598650026544,
1.01681578672332627598650026544, 2.22706506323772413245994434440, 3.22104381336452817800724765724, 4.25182304240399337595134213459, 5.22893474695512436346531564500, 6.16617490779982099980480234208, 6.49127282492691650653935217420, 8.111512982996246229039532821014, 9.138198028287935682656494669045, 10.05687926948563408464548325488
