L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·6-s − 0.999·8-s + (−1 − 1.73i)9-s + (0.866 − 1.5i)11-s + (0.866 + 1.49i)12-s − 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (1 − 1.73i)18-s + 1.73·22-s + (−0.866 + 1.49i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 − 1.5i)3-s + (−0.499 + 0.866i)4-s + 1.73·6-s − 0.999·8-s + (−1 − 1.73i)9-s + (0.866 − 1.5i)11-s + (0.866 + 1.49i)12-s − 13-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (1 − 1.73i)18-s + 1.73·22-s + (−0.866 + 1.49i)24-s + (−0.5 + 0.866i)25-s + (−0.5 − 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.907073628\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.907073628\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 19 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - 1.73T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.287073658166879233626672568524, −7.950117569594188123744083978476, −7.08552191549748514378572689350, −6.72675456629830747626953461747, −5.86073930046790263470208205370, −5.16465413115236026981141294934, −3.79522468698542921335865643707, −3.17695040322370612507516482751, −2.31791401184841372643827240235, −0.879482925229121803029960749735,
1.83085242250564752240140676837, 2.57648320240452858421920293901, 3.55395075999192802679726592975, 4.22647053183362401640695530634, 4.65597166750579192239009583212, 5.46422652052051979019744344082, 6.52217717745783540635187994040, 7.63388306770389621800443103819, 8.498674777360228994239754878690, 9.342544419452216602509877349208