L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s − 0.999·6-s + (1.5 − 2.59i)7-s − 0.999·8-s + (−0.499 − 0.866i)9-s − 0.999·10-s + 2·11-s + (−0.499 − 0.866i)12-s + (3.20 − 5.54i)13-s + 3·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (1 + 1.73i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s − 0.408·6-s + (0.566 − 0.981i)7-s − 0.353·8-s + (−0.166 − 0.288i)9-s − 0.316·10-s + 0.603·11-s + (−0.144 − 0.249i)12-s + (0.887 − 1.53i)13-s + 0.801·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.242 + 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.696 - 0.717i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.875433098\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.875433098\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (6.05 + 0.607i)T \) |
good | 7 | \( 1 + (-1.5 + 2.59i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-3.20 + 5.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.350 + 0.607i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 - 5.40T + 23T^{2} \) |
| 29 | \( 1 - 4.70T + 29T^{2} \) |
| 31 | \( 1 + 1.70T + 31T^{2} \) |
| 41 | \( 1 + (-0.701 + 1.21i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 - 3.70T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + (-4.70 - 8.14i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.70 - 2.94i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.14 + 3.72i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.35 + 2.33i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 11.7T + 73T^{2} \) |
| 79 | \( 1 + (5.55 - 9.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.649 - 1.12i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (5.70 + 9.87i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8.29T + 97T^{2} \) |
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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
−10.15292385117275358456499561390, −8.957559310997997282573865367051, −8.174895097670554213177318904779, −7.40423513079132487479862772489, −6.57118148296395674499647010766, −5.67193033445472768694989978073, −4.79243986977841072913787234119, −3.85721397265232172214004537554, −3.15068783545937049941874160077, −0.980964750474915473151526434784,
1.22028132867525132747550795302, 2.11963342434973698590390563303, 3.46636458552767120558125163374, 4.58427764105156740825410417130, 5.31059448628589767648630203124, 6.30206461455360158164205152024, 7.06113494864611259622950493717, 8.403702484755230918526270387895, 8.868050161988176318226963371373, 9.646381504990794928182263394541