L(s) = 1 | + (−0.984 + 0.173i)2-s + (1.43 − 0.974i)3-s + (0.939 − 0.342i)4-s + (−2.18 + 0.487i)5-s + (−1.24 + 1.20i)6-s + (1.28 + 0.739i)7-s + (−0.866 + 0.5i)8-s + (1.10 − 2.79i)9-s + (2.06 − 0.858i)10-s + (2.81 − 1.62i)11-s + (1.01 − 1.40i)12-s + (−3.29 − 2.76i)13-s + (−1.38 − 0.505i)14-s + (−2.65 + 2.82i)15-s + (0.766 − 0.642i)16-s + (−0.748 − 4.24i)17-s + ⋯ |
L(s) = 1 | + (−0.696 + 0.122i)2-s + (0.826 − 0.562i)3-s + (0.469 − 0.171i)4-s + (−0.975 + 0.217i)5-s + (−0.506 + 0.493i)6-s + (0.483 + 0.279i)7-s + (−0.306 + 0.176i)8-s + (0.367 − 0.930i)9-s + (0.652 − 0.271i)10-s + (0.849 − 0.490i)11-s + (0.292 − 0.405i)12-s + (−0.913 − 0.766i)13-s + (−0.371 − 0.135i)14-s + (−0.684 + 0.729i)15-s + (0.191 − 0.160i)16-s + (−0.181 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04830 - 0.634755i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04830 - 0.634755i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-1.43 + 0.974i)T \) |
| 5 | \( 1 + (2.18 - 0.487i)T \) |
| 19 | \( 1 + (-1.45 - 4.10i)T \) |
good | 7 | \( 1 + (-1.28 - 0.739i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 1.62i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.29 + 2.76i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.748 + 4.24i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-7.23 + 2.63i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.791 + 4.48i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.19 - 1.26i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 4.11T + 37T^{2} \) |
| 41 | \( 1 + (1.02 - 0.862i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.81 + 7.73i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (2.06 - 11.6i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.503 - 1.38i)T + (-40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.10 + 11.9i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-8.53 + 3.10i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (2.31 - 13.1i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (2.07 + 0.757i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (7.62 + 9.08i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-4.55 - 5.43i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.29 + 2.23i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.44 - 2.89i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-1.63 - 9.26i)T + (-91.1 + 33.1i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.50369137109623990746178496325, −9.443555361693185199277800813892, −8.681902620764936430722897555146, −7.945182745715962005347133793762, −7.30250345327868637280644755881, −6.46809428379642629962689148997, −4.95857807957601950129023357039, −3.51636350166421579602699238504, −2.55169406460618389115684714687, −0.876996959737437239337859847867,
1.57268642305142127687365590778, 3.06782917925285101482685128408, 4.19379106347386647920101537624, 4.91590158980551132700589915291, 6.97731407237178480611091819655, 7.38221661729083900053691561511, 8.509701723730672988472528039081, 9.006838915445444670061976313938, 9.835697762835462179203772810358, 10.86455544981866293937818525384