| L(s) = 1 | − 4·2-s + 24.4·3-s + 16·4-s − 97.9·6-s + 101.·7-s − 64·8-s + 356.·9-s + 291.·11-s + 391.·12-s − 1.13e3·13-s − 405.·14-s + 256·16-s + 259.·17-s − 1.42e3·18-s + 361·19-s + 2.47e3·21-s − 1.16e3·22-s − 3.90e3·23-s − 1.56e3·24-s + 4.52e3·26-s + 2.77e3·27-s + 1.62e3·28-s − 183.·29-s − 9.67e3·31-s − 1.02e3·32-s + 7.12e3·33-s − 1.03e3·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.57·3-s + 0.5·4-s − 1.11·6-s + 0.781·7-s − 0.353·8-s + 1.46·9-s + 0.725·11-s + 0.785·12-s − 1.85·13-s − 0.552·14-s + 0.250·16-s + 0.217·17-s − 1.03·18-s + 0.229·19-s + 1.22·21-s − 0.513·22-s − 1.53·23-s − 0.555·24-s + 1.31·26-s + 0.731·27-s + 0.390·28-s − 0.0404·29-s − 1.80·31-s − 0.176·32-s + 1.13·33-s − 0.153·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 4T \) |
| 5 | \( 1 \) |
| 19 | \( 1 - 361T \) |
| good | 3 | \( 1 - 24.4T + 243T^{2} \) |
| 7 | \( 1 - 101.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 291.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 1.13e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 259.T + 1.41e6T^{2} \) |
| 23 | \( 1 + 3.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 183.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.67e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 986.T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.89e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.57e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.10e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.61e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.63e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.52e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.30e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.82e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.65e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.84e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.06e5T + 8.58e9T^{2} \) |
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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
−8.895832939454158057686770398459, −8.012361691001192540172542479488, −7.62313623938788151241659552787, −6.77703521594221374947354613612, −5.33260280352784808750067404915, −4.20814470841334358701730536344, −3.22543064903643959645097831519, −2.15131089258498834453371036404, −1.63476139380048187218298034879, 0,
1.63476139380048187218298034879, 2.15131089258498834453371036404, 3.22543064903643959645097831519, 4.20814470841334358701730536344, 5.33260280352784808750067404915, 6.77703521594221374947354613612, 7.62313623938788151241659552787, 8.012361691001192540172542479488, 8.895832939454158057686770398459
