| L(s) = 1 | + (−0.908 + 1.08i)2-s + (2.01 − 1.46i)3-s + (−0.350 − 1.96i)4-s + (−1.86 − 1.23i)5-s + (−0.243 + 3.51i)6-s − 0.110i·7-s + (2.45 + 1.40i)8-s + (0.995 − 3.06i)9-s + (3.03 − 0.903i)10-s + (1.63 − 0.531i)11-s + (−3.59 − 3.45i)12-s + (1.48 − 4.56i)13-s + (0.120 + 0.100i)14-s + (−5.57 + 0.249i)15-s + (−3.75 + 1.37i)16-s + (0.269 − 0.371i)17-s + ⋯ |
| L(s) = 1 | + (−0.642 + 0.766i)2-s + (1.16 − 0.846i)3-s + (−0.175 − 0.984i)4-s + (−0.834 − 0.551i)5-s + (−0.0993 + 1.43i)6-s − 0.0419i·7-s + (0.867 + 0.497i)8-s + (0.331 − 1.02i)9-s + (0.958 − 0.285i)10-s + (0.493 − 0.160i)11-s + (−1.03 − 0.998i)12-s + (0.411 − 1.26i)13-s + (0.0321 + 0.0269i)14-s + (−1.43 + 0.0643i)15-s + (−0.938 + 0.344i)16-s + (0.0653 − 0.0899i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03123 - 0.366654i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03123 - 0.366654i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.908 - 1.08i)T \) |
| 5 | \( 1 + (1.86 + 1.23i)T \) |
| good | 3 | \( 1 + (-2.01 + 1.46i)T + (0.927 - 2.85i)T^{2} \) |
| 7 | \( 1 + 0.110iT - 7T^{2} \) |
| 11 | \( 1 + (-1.63 + 0.531i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.48 + 4.56i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (-0.269 + 0.371i)T + (-5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-2.40 + 3.30i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (6.10 - 1.98i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-5.29 - 7.28i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.87 - 2.08i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (0.500 - 1.53i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (3.51 - 10.8i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + (-1.82 - 2.51i)T + (-14.5 + 44.6i)T^{2} \) |
| 53 | \( 1 + (3.71 - 2.69i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (3.38 + 1.10i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-11.1 + 3.63i)T + (49.3 - 35.8i)T^{2} \) |
| 67 | \( 1 + (3.02 + 2.19i)T + (20.7 + 63.7i)T^{2} \) |
| 71 | \( 1 + (-9.04 + 6.57i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-9.85 + 3.20i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (4.45 - 3.23i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (9.91 + 7.20i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (2.07 + 6.38i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (3.30 + 4.54i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−12.58087336651221475557283261875, −11.39967908908890899735270397253, −10.04290569472988699208333663231, −8.852385956436491790518840263675, −8.255754777292611182553708888207, −7.60448389911693722484618257918, −6.56045182130533115033381614407, −5.00818217520173313492382754798, −3.26379975839085207353704703712, −1.20661591158370593994213780411,
2.32186630726236348771809009379, 3.73997342447074206606234216588, 4.16801127362108002593582220272, 6.76896741016659073240821719033, 8.054745580275651414908791394635, 8.625461018481902457853647251971, 9.726156723462279508947008193298, 10.31550717360339126589641428795, 11.61291576433694433102729675914, 12.09948412183806762984076764897
