| L(s) = 1 | + (−0.863 − 1.11i)2-s + (−0.872 − 1.49i)3-s + (−0.508 + 1.93i)4-s + (−2.34 + 2.20i)5-s + (−0.921 + 2.26i)6-s + (−2.77 + 0.323i)7-s + (2.60 − 1.10i)8-s + (−1.47 + 2.61i)9-s + (4.49 + 0.714i)10-s + (0.875 − 0.262i)11-s + (3.33 − 0.928i)12-s + (−2.45 + 0.143i)13-s + (2.75 + 2.82i)14-s + (5.34 + 1.57i)15-s + (−3.48 − 1.96i)16-s + (2.54 + 0.925i)17-s + ⋯ |
| L(s) = 1 | + (−0.610 − 0.791i)2-s + (−0.503 − 0.863i)3-s + (−0.254 + 0.967i)4-s + (−1.04 + 0.987i)5-s + (−0.376 + 0.926i)6-s + (−1.04 + 0.122i)7-s + (0.921 − 0.389i)8-s + (−0.492 + 0.870i)9-s + (1.42 + 0.225i)10-s + (0.263 − 0.0790i)11-s + (0.963 − 0.267i)12-s + (−0.681 + 0.0396i)13-s + (0.736 + 0.754i)14-s + (1.38 + 0.406i)15-s + (−0.870 − 0.491i)16-s + (0.617 + 0.224i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.165 + 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.372374 - 0.315143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.372374 - 0.315143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.863 + 1.11i)T \) |
| 3 | \( 1 + (0.872 + 1.49i)T \) |
| good | 5 | \( 1 + (2.34 - 2.20i)T + (0.290 - 4.99i)T^{2} \) |
| 7 | \( 1 + (2.77 - 0.323i)T + (6.81 - 1.61i)T^{2} \) |
| 11 | \( 1 + (-0.875 + 0.262i)T + (9.19 - 6.04i)T^{2} \) |
| 13 | \( 1 + (2.45 - 0.143i)T + (12.9 - 1.50i)T^{2} \) |
| 17 | \( 1 + (-2.54 - 0.925i)T + (13.0 + 10.9i)T^{2} \) |
| 19 | \( 1 + (2.26 + 6.20i)T + (-14.5 + 12.2i)T^{2} \) |
| 23 | \( 1 + (-8.66 - 1.01i)T + (22.3 + 5.30i)T^{2} \) |
| 29 | \( 1 + (2.30 + 4.59i)T + (-17.3 + 23.2i)T^{2} \) |
| 31 | \( 1 + (-0.842 + 1.13i)T + (-8.89 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-3.24 - 3.86i)T + (-6.42 + 36.4i)T^{2} \) |
| 41 | \( 1 + (-8.29 + 5.45i)T + (16.2 - 37.6i)T^{2} \) |
| 43 | \( 1 + (-0.315 + 1.33i)T + (-38.4 - 19.2i)T^{2} \) |
| 47 | \( 1 + (-2.47 - 3.32i)T + (-13.4 + 45.0i)T^{2} \) |
| 53 | \( 1 + (8.55 - 4.94i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.88 - 1.76i)T + (49.2 + 32.4i)T^{2} \) |
| 61 | \( 1 + (-1.61 - 0.694i)T + (41.8 + 44.3i)T^{2} \) |
| 67 | \( 1 + (0.313 - 0.625i)T + (-40.0 - 53.7i)T^{2} \) |
| 71 | \( 1 + (-0.130 - 0.741i)T + (-66.7 + 24.2i)T^{2} \) |
| 73 | \( 1 + (1.31 - 7.46i)T + (-68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (6.52 + 4.29i)T + (31.2 + 72.5i)T^{2} \) |
| 83 | \( 1 + (-4.77 + 7.25i)T + (-32.8 - 76.2i)T^{2} \) |
| 89 | \( 1 + (-0.806 + 4.57i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (-10.3 + 10.9i)T + (-5.64 - 96.8i)T^{2} \) |
| 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
−10.63350390855072739035858911621, −9.595024520446308718232260186072, −8.669509566638615076676808649344, −7.42308634146594845650864890677, −7.22217356660430754209939296268, −6.20797011828969982802166405984, −4.56935990840101298522951159001, −3.23489556572766005292965811021, −2.55274543874561828003256949390, −0.56726218781314557665960249062,
0.76084814694364439020728076061, 3.46438077333091577554474090594, 4.49547133073572578202971917401, 5.27494401435656738166567373132, 6.28617139827250086033619634384, 7.27102532310592489052943539747, 8.222233418057970205260905376300, 9.138230284978234612130777538319, 9.636670194429404577614297189946, 10.52614942670572616830498370643
