| L(s) = 1 | − 9.65·2-s + 61.2·4-s + 31.0·5-s − 183.·7-s − 281.·8-s − 299.·10-s − 873.·13-s + 1.77e3·14-s + 763.·16-s + 775.·17-s − 1.49e3·19-s + 1.90e3·20-s − 1.30e3·23-s − 2.15e3·25-s + 8.43e3·26-s − 1.12e4·28-s − 485.·29-s + 7.55e3·31-s + 1.65e3·32-s − 7.48e3·34-s − 5.70e3·35-s + 1.16e4·37-s + 1.44e4·38-s − 8.76e3·40-s + 8.26e3·41-s + 1.74e3·43-s + 1.25e4·46-s + ⋯ |
| L(s) = 1 | − 1.70·2-s + 1.91·4-s + 0.555·5-s − 1.41·7-s − 1.55·8-s − 0.948·10-s − 1.43·13-s + 2.41·14-s + 0.745·16-s + 0.650·17-s − 0.951·19-s + 1.06·20-s − 0.513·23-s − 0.691·25-s + 2.44·26-s − 2.70·28-s − 0.107·29-s + 1.41·31-s + 0.285·32-s − 1.11·34-s − 0.786·35-s + 1.40·37-s + 1.62·38-s − 0.865·40-s + 0.768·41-s + 0.144·43-s + 0.876·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 9.65T + 32T^{2} \) |
| 5 | \( 1 - 31.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 183.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 873.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 775.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.49e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.30e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 485.T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.55e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.16e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.74e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.66e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.40e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.40e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 7.05e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.60e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.18e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.62e3T + 3.93e9T^{2} \) |
| 89 | \( 1 + 9.33e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.19e4T + 8.58e9T^{2} \) |
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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
−8.992538617340413556165906699604, −7.953560169250351855318936528787, −7.28896429179541547653676532302, −6.42383071257771229856981385006, −5.80509002640716085421825245914, −4.25581457702037421217542260946, −2.76757903183241888279135359213, −2.22464692528771889903345272551, −0.847588189911438492264876310240, 0,
0.847588189911438492264876310240, 2.22464692528771889903345272551, 2.76757903183241888279135359213, 4.25581457702037421217542260946, 5.80509002640716085421825245914, 6.42383071257771229856981385006, 7.28896429179541547653676532302, 7.953560169250351855318936528787, 8.992538617340413556165906699604
