L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (0.173 + 0.300i)11-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)32-s + (−0.766 − 0.642i)34-s + (0.5 + 0.866i)38-s + (−0.266 − 1.50i)41-s + (−0.766 + 0.642i)43-s + (−0.0603 + 0.342i)44-s + ⋯ |
L(s) = 1 | + (0.939 + 0.342i)2-s + (0.766 + 0.642i)4-s + (0.500 + 0.866i)8-s + (0.173 + 0.300i)11-s + (0.173 + 0.984i)16-s + (−0.939 − 0.342i)17-s + (0.766 + 0.642i)19-s + (0.0603 + 0.342i)22-s + (0.173 − 0.984i)25-s + (−0.173 + 0.984i)32-s + (−0.766 − 0.642i)34-s + (0.5 + 0.866i)38-s + (−0.266 − 1.50i)41-s + (−0.766 + 0.642i)43-s + (−0.0603 + 0.342i)44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.672 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.904553860\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.904553860\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.939 - 0.342i)T \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-0.766 - 0.642i)T \) |
good | 5 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.173 - 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 17 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (0.266 + 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 43 | \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 47 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 53 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 59 | \( 1 + (-0.326 - 0.118i)T + (0.766 + 0.642i)T^{2} \) |
| 61 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 67 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 73 | \( 1 + (-0.0603 - 0.342i)T + (-0.939 + 0.342i)T^{2} \) |
| 79 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 83 | \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.173 + 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 97 | \( 1 + (1.43 + 0.524i)T + (0.766 + 0.642i)T^{2} \) |
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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
−10.00668967021357197950152520171, −8.929333028571894099971059184639, −8.132989832654868891461726244412, −7.24161792724496981537237931069, −6.58995517168230179449110000294, −5.69836785865108672941675352813, −4.82474859965014723984427307759, −4.02334268265414628921121604209, −3.00082475588478984637833464627, −1.89424456873575858825120713011,
1.44625138924848356055359883944, 2.71659461309361275623900224654, 3.59426921854885338489579199497, 4.60591103935087721741175043268, 5.34917796674674607365311047076, 6.32634408214956810603503923595, 6.97522334131577793483354100974, 7.940482120776892392742541174260, 9.058693951261213878468494047946, 9.748575149841909369467442713333