L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−0.363 + 2.20i)5-s + 0.999·6-s + (−0.268 + 0.154i)7-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.41 + 1.72i)10-s + 4.57·11-s + (0.866 + 0.499i)12-s + (−3.32 + 1.91i)13-s − 0.309·14-s + (0.788 + 2.09i)15-s + (−0.5 + 0.866i)16-s + (2.86 + 1.65i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.162 + 0.986i)5-s + 0.408·6-s + (−0.101 + 0.0585i)7-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.448 + 0.546i)10-s + 1.37·11-s + (0.249 + 0.144i)12-s + (−0.921 + 0.531i)13-s − 0.0827·14-s + (0.203 + 0.540i)15-s + (−0.125 + 0.216i)16-s + (0.695 + 0.401i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.675821878\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.675821878\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (0.363 - 2.20i)T \) |
| 37 | \( 1 + (-6.02 - 0.841i)T \) |
good | 7 | \( 1 + (0.268 - 0.154i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 + (3.32 - 1.91i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.86 - 1.65i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.688 + 1.19i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 0.690T + 29T^{2} \) |
| 31 | \( 1 - 2.73T + 31T^{2} \) |
| 41 | \( 1 + (-0.175 - 0.303i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 5.04iT - 43T^{2} \) |
| 47 | \( 1 - 3.77iT - 47T^{2} \) |
| 53 | \( 1 + (11.8 + 6.84i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.77 + 10.0i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.37 - 4.12i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.13 + 2.38i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.67 + 11.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 13.3iT - 73T^{2} \) |
| 79 | \( 1 + (-7.63 - 13.2i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.66 - 1.53i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.47 + 9.48i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.990iT - 97T^{2} \) |
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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
−9.768711722291545339516768739021, −9.350115458801346876047557198830, −8.055265178167595738548019395727, −7.46537987666938803619374832968, −6.60053863745705004359337128040, −6.12629414239461845880086043381, −4.72046185365047454741017032735, −3.74022074650142729830085348633, −2.99747486924289552050703249324, −1.78012587502798277123978827998,
0.971800970266334830175940904546, 2.34677235236564180915173248591, 3.55035881027290393319799840180, 4.35034469765501423719274682718, 5.08992279552439702353974082495, 6.09801409133684299286082575805, 7.17309205350868967198590225627, 8.128559652109132246293761128025, 8.960755633838122593291028972494, 9.694436957101812555793924479831