L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.266 + 1.50i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)16-s + (0.939 − 1.62i)17-s + 0.999·18-s + (−0.173 − 0.300i)19-s + (0.266 − 1.50i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯ |
L(s) = 1 | + (−0.939 − 0.342i)2-s + (0.173 + 0.984i)3-s + (0.766 + 0.642i)4-s + (0.173 − 0.984i)6-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (0.266 + 1.50i)11-s + (−0.5 + 0.866i)12-s + (0.173 + 0.984i)16-s + (0.939 − 1.62i)17-s + 0.999·18-s + (−0.173 − 0.300i)19-s + (0.266 − 1.50i)22-s + (0.766 − 0.642i)24-s + (−0.939 − 0.342i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.727 - 0.686i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5069317621\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5069317621\) |
\(L(1)\) |
|
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 | \( 1 + (-0.173 - 0.984i)T \) |
good | 5 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 11 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 29 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 31 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.326 - 0.118i)T + (0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.326 + 1.85i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \) |
| 61 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 67 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 71 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.939 - 1.62i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.939 - 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.266 - 1.50i)T + (-0.939 + 0.342i)T^{2} \) |
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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
−12.16673406637422495019177931662, −11.63939972661834752743021834987, −10.39022260009512089582157799522, −9.765531946411498251344924338403, −9.087986225846525406667411900895, −7.87354725130302796243081668553, −6.89665279786183129393175293035, −5.19134054502797220968515953638, −3.83801659828596263560765615994, −2.40222102885017417011513209495,
1.50292278577570919072792984455, 3.26958848221300922261022230232, 5.82766409258450834246922328554, 6.28765311599362488319783499160, 7.73169680743094807641806986817, 8.263283939764319724125817982648, 9.220764964272969675686154178365, 10.51065905503647610025758177583, 11.39380713297973286218062862119, 12.29254082495383292127253192517