| L(s) = 1 | + (0.766 + 0.642i)3-s + (3.02 − 1.10i)5-s + (1.81 − 1.04i)7-s + (0.173 + 0.984i)9-s + (−1.32 − 0.762i)11-s + (3.47 + 4.13i)13-s + (3.02 + 1.10i)15-s + (−0.0697 + 0.395i)17-s + (−1.70 − 4.01i)19-s + (2.06 + 0.364i)21-s + (−0.994 + 2.73i)23-s + (4.11 − 3.45i)25-s + (−0.500 + 0.866i)27-s + (−0.725 + 0.127i)29-s + (−2.72 − 4.72i)31-s + ⋯ |
| L(s) = 1 | + (0.442 + 0.371i)3-s + (1.35 − 0.492i)5-s + (0.686 − 0.396i)7-s + (0.0578 + 0.328i)9-s + (−0.398 − 0.230i)11-s + (0.963 + 1.14i)13-s + (0.781 + 0.284i)15-s + (−0.0169 + 0.0958i)17-s + (−0.391 − 0.920i)19-s + (0.450 + 0.0794i)21-s + (−0.207 + 0.569i)23-s + (0.823 − 0.691i)25-s + (−0.0962 + 0.166i)27-s + (−0.134 + 0.0237i)29-s + (−0.490 − 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.48177 + 0.0368573i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.48177 + 0.0368573i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 19 | \( 1 + (1.70 + 4.01i)T \) |
| good | 5 | \( 1 + (-3.02 + 1.10i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-1.81 + 1.04i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (1.32 + 0.762i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.47 - 4.13i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.0697 - 0.395i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (0.994 - 2.73i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.725 - 0.127i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.72 + 4.72i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 3.91iT - 37T^{2} \) |
| 41 | \( 1 + (-3.82 + 4.55i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-2.67 - 7.35i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (9.18 - 1.61i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (0.116 - 0.319i)T + (-40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.81 + 10.3i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (2.45 + 0.892i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-1.74 - 9.89i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.09 - 0.397i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (-4.32 - 3.62i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-8.16 - 6.84i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (11.4 - 6.63i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.157 + 0.188i)T + (-15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-15.3 - 2.69i)T + (91.1 + 33.1i)T^{2} \) |
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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
−9.872605631385067109030837170685, −9.289437531001941561319884635665, −8.636235543606131181674275753222, −7.71723938174702831349549621900, −6.55177938536857487244265949165, −5.68159426586941296320055914863, −4.77973076745853641968958410176, −3.88168754235883336018380947682, −2.37343005870584829624988591520, −1.45192477932620668223688173672,
1.52030168158171472429263928613, 2.41011302571557769920629873190, 3.45698682304435121843157124673, 5.01320331514011133897458458001, 5.86488642691079082944655526014, 6.49691825754537674341920506365, 7.70753520231755896339412172271, 8.388700953646087947496537547451, 9.188077179213055164958758455371, 10.30322715516156117992152797371
