| L(s) = 1 | + (−0.0523 + 0.998i)2-s + (−0.571 − 1.48i)3-s + (−0.994 − 0.104i)4-s + (−0.530 − 2.17i)5-s + (1.51 − 0.493i)6-s + (2.02 − 1.70i)7-s + (0.156 − 0.987i)8-s + (0.336 − 0.303i)9-s + (2.19 − 0.415i)10-s + (−2.46 − 2.21i)11-s + (0.413 + 1.54i)12-s + (−0.930 + 0.473i)13-s + (1.59 + 2.11i)14-s + (−2.93 + 2.03i)15-s + (0.978 + 0.207i)16-s + (0.257 + 4.90i)17-s + ⋯ |
| L(s) = 1 | + (−0.0370 + 0.706i)2-s + (−0.330 − 0.860i)3-s + (−0.497 − 0.0522i)4-s + (−0.237 − 0.971i)5-s + (0.619 − 0.201i)6-s + (0.766 − 0.642i)7-s + (0.0553 − 0.349i)8-s + (0.112 − 0.101i)9-s + (0.694 − 0.131i)10-s + (−0.743 − 0.668i)11-s + (0.119 + 0.445i)12-s + (−0.257 + 0.131i)13-s + (0.425 + 0.564i)14-s + (−0.757 + 0.524i)15-s + (0.244 + 0.0519i)16-s + (0.0623 + 1.19i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 770 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 + 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.421197 - 0.829899i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.421197 - 0.829899i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.0523 - 0.998i)T \) |
| 5 | \( 1 + (0.530 + 2.17i)T \) |
| 7 | \( 1 + (-2.02 + 1.70i)T \) |
| 11 | \( 1 + (2.46 + 2.21i)T \) |
| good | 3 | \( 1 + (0.571 + 1.48i)T + (-2.22 + 2.00i)T^{2} \) |
| 13 | \( 1 + (0.930 - 0.473i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.257 - 4.90i)T + (-16.9 + 1.77i)T^{2} \) |
| 19 | \( 1 + (0.153 + 1.45i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (1.39 + 5.18i)T + (-19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (0.541 - 0.745i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.13 - 10.0i)T + (-28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (1.77 + 0.682i)T + (27.4 + 24.7i)T^{2} \) |
| 41 | \( 1 + (2.93 + 4.03i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + (-5.11 + 5.11i)T - 43iT^{2} \) |
| 47 | \( 1 + (8.64 + 7.00i)T + (9.77 + 45.9i)T^{2} \) |
| 53 | \( 1 + (-2.46 + 3.79i)T + (-21.5 - 48.4i)T^{2} \) |
| 59 | \( 1 + (0.358 - 3.41i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (-1.60 + 7.52i)T + (-55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (0.648 - 2.42i)T + (-58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-1.76 - 5.42i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (6.83 - 5.53i)T + (15.1 - 71.4i)T^{2} \) |
| 79 | \( 1 + (-1.80 + 1.62i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-4.13 + 8.11i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (4.16 - 7.21i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.592 + 1.16i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.05074706117951235025749489417, −8.595191390455367271178412592926, −8.356986865523592839432404066089, −7.39488102180414577492255905941, −6.66009285306177153719039008116, −5.59684314819055529513561675960, −4.80031748241094993316067659971, −3.79787249199176994749578710781, −1.70664424324563069810840267191, −0.50184078768705429375917034487,
2.05314322143680421391562272501, 3.02299568653731031110702244355, 4.28775623928922331486907390782, 4.99838907695275510617671856168, 5.92445980653250389892947528026, 7.50043111120767218174443940299, 7.904404657292044014095874421525, 9.415927328265039948111684040907, 9.863400111713194146962088863819, 10.63163058834279832628563519743
