| L(s) = 1 | + (0.927 − 1.06i)2-s + (−1.07 − 1.35i)3-s + (−0.280 − 1.98i)4-s + (0.0465 + 0.127i)5-s + (−2.44 − 0.113i)6-s + (−0.252 − 1.42i)7-s + (−2.37 − 1.53i)8-s + (−0.693 + 2.91i)9-s + (0.179 + 0.0688i)10-s + (0.771 − 2.11i)11-s + (−2.38 + 2.50i)12-s + (−0.634 + 0.755i)13-s + (−1.76 − 1.05i)14-s + (0.123 − 0.200i)15-s + (−3.84 + 1.11i)16-s + (−0.439 + 0.760i)17-s + ⋯ |
| L(s) = 1 | + (0.655 − 0.755i)2-s + (−0.619 − 0.784i)3-s + (−0.140 − 0.990i)4-s + (0.0208 + 0.0572i)5-s + (−0.998 − 0.0461i)6-s + (−0.0952 − 0.540i)7-s + (−0.839 − 0.542i)8-s + (−0.231 + 0.972i)9-s + (0.0568 + 0.0217i)10-s + (0.232 − 0.638i)11-s + (−0.689 + 0.724i)12-s + (−0.175 + 0.209i)13-s + (−0.470 − 0.282i)14-s + (0.0319 − 0.0518i)15-s + (−0.960 + 0.278i)16-s + (−0.106 + 0.184i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.815 + 0.579i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.388986 - 1.21911i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.388986 - 1.21911i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.927 + 1.06i)T \) |
| 3 | \( 1 + (1.07 + 1.35i)T \) |
| good | 5 | \( 1 + (-0.0465 - 0.127i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.252 + 1.42i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.771 + 2.11i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (0.634 - 0.755i)T + (-2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (0.439 - 0.760i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.20 + 3.00i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.748 - 4.24i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.146 - 0.174i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.15 + 6.56i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-9.25 - 5.34i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.19 + 1.00i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.11 + 3.05i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.36 - 7.74i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 - 5.08iT - 53T^{2} \) |
| 59 | \( 1 + (3.15 + 8.68i)T + (-45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (12.3 - 2.18i)T + (57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.90 - 7.03i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (5.93 - 10.2i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.88 - 8.45i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.49 + 4.60i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (0.867 + 1.03i)T + (-14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (3.19 + 5.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.779 - 0.283i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.80894909662661425584006964441, −11.30446151683726894491825567629, −10.34866272586761435825517884316, −9.226706590404576079157782928971, −7.68232300206355486881968519545, −6.55286241702356702072639032544, −5.62046142687216271523756077637, −4.38787112731426758773477413503, −2.80555037856822927831874298558, −1.03998502751080578701561834990,
3.11514432210996672204711142883, 4.46889604701487972448573396211, 5.35939505527560103263591279131, 6.31929750600901638038035898425, 7.45816916115926829305637621630, 8.819878360997525306760086772589, 9.651800678500531331526942730133, 10.90326135215660738368088668551, 12.10679160721475739866253701739, 12.42703067176042177617669778837
