| L(s) = 1 | + (1.25 + 0.655i)2-s + (0.741 + 1.35i)3-s + (1.14 + 1.64i)4-s + (−2.06 − 0.849i)5-s + (0.0391 + 2.18i)6-s + (1.32 + 1.76i)7-s + (0.352 + 2.80i)8-s + (0.327 − 0.509i)9-s + (−2.03 − 2.42i)10-s + (−1.69 + 0.776i)11-s + (−1.38 + 2.76i)12-s + (3.54 + 2.65i)13-s + (0.498 + 3.07i)14-s + (−0.380 − 3.43i)15-s + (−1.39 + 3.74i)16-s + (−3.84 + 0.274i)17-s + ⋯ |
| L(s) = 1 | + (0.886 + 0.463i)2-s + (0.428 + 0.784i)3-s + (0.570 + 0.821i)4-s + (−0.925 − 0.379i)5-s + (0.0159 + 0.893i)6-s + (0.499 + 0.667i)7-s + (0.124 + 0.992i)8-s + (0.109 − 0.169i)9-s + (−0.643 − 0.765i)10-s + (−0.512 + 0.234i)11-s + (−0.399 + 0.798i)12-s + (0.983 + 0.736i)13-s + (0.133 + 0.822i)14-s + (−0.0982 − 0.888i)15-s + (−0.349 + 0.936i)16-s + (−0.931 + 0.0666i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.244 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.48019 + 1.89979i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.48019 + 1.89979i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.25 - 0.655i)T \) |
| 5 | \( 1 + (2.06 + 0.849i)T \) |
| 23 | \( 1 + (-2.50 + 4.09i)T \) |
| good | 3 | \( 1 + (-0.741 - 1.35i)T + (-1.62 + 2.52i)T^{2} \) |
| 7 | \( 1 + (-1.32 - 1.76i)T + (-1.97 + 6.71i)T^{2} \) |
| 11 | \( 1 + (1.69 - 0.776i)T + (7.20 - 8.31i)T^{2} \) |
| 13 | \( 1 + (-3.54 - 2.65i)T + (3.66 + 12.4i)T^{2} \) |
| 17 | \( 1 + (3.84 - 0.274i)T + (16.8 - 2.41i)T^{2} \) |
| 19 | \( 1 + (0.767 + 0.885i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (6.13 + 5.31i)T + (4.12 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.584 - 1.98i)T + (-26.0 + 16.7i)T^{2} \) |
| 37 | \( 1 + (1.66 + 7.65i)T + (-33.6 + 15.3i)T^{2} \) |
| 41 | \( 1 + (-8.23 + 5.29i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-3.66 - 6.71i)T + (-23.2 + 36.1i)T^{2} \) |
| 47 | \( 1 + (-6.69 + 6.69i)T - 47iT^{2} \) |
| 53 | \( 1 + (0.345 + 0.461i)T + (-14.9 + 50.8i)T^{2} \) |
| 59 | \( 1 + (-0.146 - 1.01i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-1.92 + 0.564i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (1.52 + 4.09i)T + (-50.6 + 43.8i)T^{2} \) |
| 71 | \( 1 + (3.65 + 1.66i)T + (46.4 + 53.6i)T^{2} \) |
| 73 | \( 1 + (4.31 + 0.308i)T + (72.2 + 10.3i)T^{2} \) |
| 79 | \( 1 + (-0.689 - 4.79i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (-2.77 - 12.7i)T + (-75.4 + 34.4i)T^{2} \) |
| 89 | \( 1 + (2.93 - 9.98i)T + (-74.8 - 48.1i)T^{2} \) |
| 97 | \( 1 + (-7.50 - 1.63i)T + (88.2 + 40.2i)T^{2} \) |
| 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
−11.33284418997606269304766140498, −10.79221554942602484638895079409, −9.054646235612390363822858358011, −8.710120657817913639018130289829, −7.66579071406284085132449288639, −6.62385302797301219556115446884, −5.39024060259268472416362036951, −4.33868975436370454533654939674, −3.88392540233782545912201070398, −2.42530904124964426442119631897,
1.23889612399629246557179243129, 2.72977336647859507548503526689, 3.77377412122073626844903093189, 4.79370630845717689466486186376, 6.09954987582141495603489156917, 7.28047185945995872557494903462, 7.69256991077867394732893136116, 8.825095752204768045714853813157, 10.49162499654128145696353270108, 10.90521548100829900893518998485
