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 + (0.499 − 0.866i)6-s − 4.35·7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−0.499 + 0.866i)10-s − 3·11-s + 0.999·12-s + (2 − 3.46i)13-s + (−2.17 − 3.77i)14-s + (0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (−1.67 − 2.90i)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 + (0.204 − 0.353i)6-s − 1.64·7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.158 + 0.273i)10-s − 0.904·11-s + 0.288·12-s + (0.554 − 0.960i)13-s + (−0.582 − 1.00i)14-s + (0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (−0.407 − 0.705i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.671 + 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0748582 - 0.168909i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0748582 - 0.168909i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (2.17 + 3.77i)T \) |
good | 7 | \( 1 + 4.35T + 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + (-2 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.67 + 2.90i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.5 + 2.59i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (4.67 - 8.10i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 10.7T + 31T^{2} \) |
| 37 | \( 1 + 7T + 37T^{2} \) |
| 41 | \( 1 + (-3.17 - 5.50i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-5 - 8.66i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6.53 + 11.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (6.35 + 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.67 - 4.64i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (5.67 - 9.83i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-5.03 - 8.72i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.32 + 4.01i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.35 - 4.08i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + (0.179 - 0.310i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.03 - 12.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.52325981500705287935842160499, −9.438028183305060786889979548674, −8.596536613407218102778969741490, −7.29974219334851439077275392342, −6.84818255786111597811264061326, −5.90690892526157251968392702098, −5.15533378634657451355223621147, −3.49853706084626224085784772319, −2.65727370149076214949415935096, −0.089032823645111286563519576607,
2.08740078002895903077963268781, 3.53901899644976064063881312067, 4.18111194001153863925359685124, 5.67232768690701720642479908622, 6.08073476862071474196335228095, 7.38241732797961059606230005261, 9.001306323609548345369321401063, 9.293337254533572796871403405930, 10.41860039317908647866050457393, 10.78597656144941086213255137445