| L(s) = 1 | + (−0.457 − 0.626i)2-s + (−0.119 + 2.24i)3-s + (0.426 − 1.33i)4-s + (3.52 − 0.0752i)5-s + (1.45 − 0.950i)6-s + (1.34 + 2.27i)7-s + (−2.50 + 0.830i)8-s + (−2.02 − 0.217i)9-s + (−1.65 − 2.17i)10-s + (0.184 − 1.56i)11-s + (2.93 + 1.11i)12-s + (−0.914 − 1.75i)13-s + (0.814 − 1.88i)14-s + (−0.253 + 7.89i)15-s + (−0.610 − 0.435i)16-s + (1.65 − 0.395i)17-s + ⋯ |
| L(s) = 1 | + (−0.323 − 0.443i)2-s + (−0.0692 + 1.29i)3-s + (0.213 − 0.665i)4-s + (1.57 − 0.0336i)5-s + (0.595 − 0.387i)6-s + (0.507 + 0.861i)7-s + (−0.884 + 0.293i)8-s + (−0.675 − 0.0724i)9-s + (−0.524 − 0.686i)10-s + (0.0555 − 0.470i)11-s + (0.846 + 0.321i)12-s + (−0.253 − 0.486i)13-s + (0.217 − 0.503i)14-s + (−0.0654 + 2.03i)15-s + (−0.152 − 0.108i)16-s + (0.400 − 0.0959i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 343 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48538 + 0.223724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48538 + 0.223724i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-1.34 - 2.27i)T \) |
| good | 2 | \( 1 + (0.457 + 0.626i)T + (-0.609 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.119 - 2.24i)T + (-2.98 - 0.319i)T^{2} \) |
| 5 | \( 1 + (-3.52 + 0.0752i)T + (4.99 - 0.213i)T^{2} \) |
| 11 | \( 1 + (-0.184 + 1.56i)T + (-10.6 - 2.56i)T^{2} \) |
| 13 | \( 1 + (0.914 + 1.75i)T + (-7.43 + 10.6i)T^{2} \) |
| 17 | \( 1 + (-1.65 + 0.395i)T + (15.1 - 7.70i)T^{2} \) |
| 19 | \( 1 + (0.315 - 0.545i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (6.75 - 5.27i)T + (5.59 - 22.3i)T^{2} \) |
| 29 | \( 1 + (-6.44 + 2.61i)T + (20.8 - 20.1i)T^{2} \) |
| 31 | \( 1 + (-4.35 + 4.04i)T + (2.31 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-5.34 - 5.63i)T + (-1.97 + 36.9i)T^{2} \) |
| 41 | \( 1 + (-2.52 + 6.86i)T + (-31.2 - 26.5i)T^{2} \) |
| 43 | \( 1 + (6.02 + 2.00i)T + (34.4 + 25.7i)T^{2} \) |
| 47 | \( 1 + (6.98 + 3.74i)T + (26.0 + 39.1i)T^{2} \) |
| 53 | \( 1 + (6.25 + 3.69i)T + (25.5 + 46.4i)T^{2} \) |
| 59 | \( 1 + (2.81 - 3.37i)T + (-10.6 - 58.0i)T^{2} \) |
| 61 | \( 1 + (5.76 - 0.494i)T + (60.1 - 10.3i)T^{2} \) |
| 67 | \( 1 + (-4.10 + 1.26i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (14.3 + 5.81i)T + (51.0 + 49.3i)T^{2} \) |
| 73 | \( 1 + (-0.000803 + 0.000430i)T + (40.4 - 60.7i)T^{2} \) |
| 79 | \( 1 + (-7.15 + 1.07i)T + (75.4 - 23.2i)T^{2} \) |
| 83 | \( 1 + (0.570 - 0.111i)T + (76.9 - 31.1i)T^{2} \) |
| 89 | \( 1 + (4.71 - 4.76i)T + (-0.951 - 88.9i)T^{2} \) |
| 97 | \( 1 + (1.17 + 5.13i)T + (-87.3 + 42.0i)T^{2} \) |
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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
−11.30146071456306291995298659547, −10.24703206075521859011130386848, −9.906824301972560360569298940310, −9.290743759266691345043596156332, −8.262759564472064534841898277687, −6.16113214386458590536048792002, −5.65975409514845825643814240032, −4.80831796603951195349340330061, −2.95153979415355097940092765287, −1.76169367485336136341987667283,
1.48621833539102826883491193145, 2.60071541850725878699649028059, 4.57932322691615488859343249032, 6.28457473515613819541641700143, 6.58391846951951621205984008480, 7.62093786111506609863941831199, 8.333905535851463375362526450798, 9.576624814119712252274890576546, 10.39872397652789899612774511183, 11.73043044249482254605563522102
