| L(s) = 1 | + (−0.407 + 2.22i)2-s + (−1.46 + 0.924i)3-s + (−2.90 − 1.10i)4-s + (3.11 + 0.188i)5-s + (−1.45 − 3.63i)6-s + (1.33 + 0.806i)7-s + (1.29 − 2.13i)8-s + (1.28 − 2.70i)9-s + (−1.68 + 6.83i)10-s + (0.662 + 3.61i)11-s + (5.27 − 1.07i)12-s + (−0.793 + 3.51i)13-s + (−2.33 + 2.63i)14-s + (−4.72 + 2.60i)15-s + (−0.421 − 0.373i)16-s + (−1.84 + 0.453i)17-s + ⋯ |
| L(s) = 1 | + (−0.288 + 1.57i)2-s + (−0.845 + 0.533i)3-s + (−1.45 − 0.550i)4-s + (1.39 + 0.0841i)5-s + (−0.595 − 1.48i)6-s + (0.504 + 0.304i)7-s + (0.457 − 0.756i)8-s + (0.429 − 0.902i)9-s + (−0.532 + 2.16i)10-s + (0.199 + 1.09i)11-s + (1.52 − 0.309i)12-s + (−0.220 + 0.975i)13-s + (−0.624 + 0.704i)14-s + (−1.22 + 0.671i)15-s + (−0.105 − 0.0934i)16-s + (−0.446 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.932 + 0.362i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.196021 - 1.04567i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.196021 - 1.04567i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.46 - 0.924i)T \) |
| 13 | \( 1 + (0.793 - 3.51i)T \) |
| good | 2 | \( 1 + (0.407 - 2.22i)T + (-1.87 - 0.709i)T^{2} \) |
| 5 | \( 1 + (-3.11 - 0.188i)T + (4.96 + 0.602i)T^{2} \) |
| 7 | \( 1 + (-1.33 - 0.806i)T + (3.25 + 6.19i)T^{2} \) |
| 11 | \( 1 + (-0.662 - 3.61i)T + (-10.2 + 3.90i)T^{2} \) |
| 17 | \( 1 + (1.84 - 0.453i)T + (15.0 - 7.90i)T^{2} \) |
| 19 | \( 1 + (0.649 - 0.649i)T - 19iT^{2} \) |
| 23 | \( 1 - 2.86T + 23T^{2} \) |
| 29 | \( 1 + (3.34 - 2.30i)T + (10.2 - 27.1i)T^{2} \) |
| 31 | \( 1 + (0.107 + 0.0840i)T + (7.41 + 30.0i)T^{2} \) |
| 37 | \( 1 + (-5.54 + 7.07i)T + (-8.85 - 35.9i)T^{2} \) |
| 41 | \( 1 + (-0.0661 - 0.212i)T + (-33.7 + 23.2i)T^{2} \) |
| 43 | \( 1 + (7.19 + 0.873i)T + (41.7 + 10.2i)T^{2} \) |
| 47 | \( 1 + (-8.81 - 3.96i)T + (31.1 + 35.1i)T^{2} \) |
| 53 | \( 1 + (0.861 + 3.49i)T + (-46.9 + 24.6i)T^{2} \) |
| 59 | \( 1 + (0.600 - 9.93i)T + (-58.5 - 7.11i)T^{2} \) |
| 61 | \( 1 + (11.7 + 2.89i)T + (54.0 + 28.3i)T^{2} \) |
| 67 | \( 1 + (8.03 + 3.61i)T + (44.4 + 50.1i)T^{2} \) |
| 71 | \( 1 + (0.388 + 1.24i)T + (-58.4 + 40.3i)T^{2} \) |
| 73 | \( 1 + (-11.1 + 2.05i)T + (68.2 - 25.8i)T^{2} \) |
| 79 | \( 1 + (-2.60 - 6.86i)T + (-59.1 + 52.3i)T^{2} \) |
| 83 | \( 1 + (2.82 + 0.881i)T + (68.3 + 47.1i)T^{2} \) |
| 89 | \( 1 + (-5.66 + 5.66i)T - 89iT^{2} \) |
| 97 | \( 1 + (-14.7 + 0.889i)T + (96.2 - 11.6i)T^{2} \) |
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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
−11.23794892326096850885744361867, −10.20669023222021872591400782080, −9.304917019546855615139523102489, −9.029605423494025925825557590929, −7.45941872181702330690894715934, −6.66289969157592688881411409290, −5.98353209729641518053817954602, −5.14859184220220417074540886711, −4.42758212967041098649623149858, −1.94336134212081354593227417012,
0.807230606349607590793112763761, 1.85560902188894219971647340451, 2.98640292748438845640136455481, 4.62445794116926198897872519046, 5.64050113202944618662560437724, 6.50065357480861890492514207488, 7.930834051504066559352159577296, 8.993542447998280174690356499371, 9.895127939779239552761801721615, 10.64860525462640570052902681670
