L(s) = 1 | + (0.5 − 0.866i)3-s − 4·7-s + (1 + 1.73i)9-s − 3·11-s + (1 + 1.73i)13-s + (3 − 5.19i)17-s + (3.5 + 2.59i)19-s + (−2 + 3.46i)21-s + (3 + 5.19i)23-s + (2.5 + 4.33i)25-s + 5·27-s + 2·31-s + (−1.5 + 2.59i)33-s + 10·37-s + 1.99·39-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s − 1.51·7-s + (0.333 + 0.577i)9-s − 0.904·11-s + (0.277 + 0.480i)13-s + (0.727 − 1.26i)17-s + (0.802 + 0.596i)19-s + (−0.436 + 0.755i)21-s + (0.625 + 1.08i)23-s + (0.5 + 0.866i)25-s + 0.962·27-s + 0.359·31-s + (−0.261 + 0.452i)33-s + 1.64·37-s + 0.320·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 - 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.411911110\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.411911110\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (-3.5 - 2.59i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + 4T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-3 + 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 2T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2 - 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (4.5 - 7.79i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (3.5 + 6.06i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-3 + 5.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3T + 83T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (8.5 - 14.7i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.648076560967476764093598862214, −9.248943792876180282106526506103, −7.900225712865470621351691421551, −7.44424171897927928764187474818, −6.63394717743279723533903728594, −5.65396617700276012431969186488, −4.76468475562860726325209260769, −3.29096627504812227289805055781, −2.78225318144012742940223962659, −1.21483248383309752623603914887,
0.66564732616554931686820538104, 2.72847360440711388628494488929, 3.35407486990990152512571686815, 4.29876058044247304345007542289, 5.48198430761843276797236102550, 6.35575204356113544667431827066, 7.05006633339860145427753795746, 8.204052810644402587360936798804, 8.888678574573461179085764885642, 9.892050146137477846198667778016