| L(s) = 1 | + (−1.51 − 0.848i)3-s − i·5-s − 3.02i·7-s + (1.56 + 2.56i)9-s + 1.32·11-s + 5.12·13-s + (−0.848 + 1.51i)15-s − 2i·17-s − 1.32i·19-s + (−2.56 + 4.56i)21-s − 0.371·23-s − 25-s + (−0.185 − 5.19i)27-s + 3.12i·29-s − 4.71i·31-s + ⋯ |
| L(s) = 1 | + (−0.871 − 0.489i)3-s − 0.447i·5-s − 1.14i·7-s + (0.520 + 0.853i)9-s + 0.399·11-s + 1.42·13-s + (−0.218 + 0.389i)15-s − 0.485i·17-s − 0.303i·19-s + (−0.558 + 0.995i)21-s − 0.0775·23-s − 0.200·25-s + (−0.0357 − 0.999i)27-s + 0.579i·29-s − 0.847i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.556374 - 0.950529i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.556374 - 0.950529i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.51 + 0.848i)T \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 3.02iT - 7T^{2} \) |
| 11 | \( 1 - 1.32T + 11T^{2} \) |
| 13 | \( 1 - 5.12T + 13T^{2} \) |
| 17 | \( 1 + 2iT - 17T^{2} \) |
| 19 | \( 1 + 1.32iT - 19T^{2} \) |
| 23 | \( 1 + 0.371T + 23T^{2} \) |
| 29 | \( 1 - 3.12iT - 29T^{2} \) |
| 31 | \( 1 + 4.71iT - 31T^{2} \) |
| 37 | \( 1 + 5.12T + 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 7.73iT - 43T^{2} \) |
| 47 | \( 1 + 3.02T + 47T^{2} \) |
| 53 | \( 1 + 12.2iT - 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 3.12T + 61T^{2} \) |
| 67 | \( 1 - 4.34iT - 67T^{2} \) |
| 71 | \( 1 + 3.39T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 8.10iT - 79T^{2} \) |
| 83 | \( 1 - 15.1T + 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−9.912755174035961761726393257560, −8.864205803624170666182629854656, −7.952227819775263400436554917516, −7.07615078308633153742224006634, −6.41468879451325239542999161246, −5.44679776594268641586739236877, −4.48205159742406791531800551973, −3.57754742081845160691364355775, −1.67414901358334986745375728106, −0.63019049406000942639412338643,
1.52509345949167648296592625454, 3.13867065856661612159732177044, 4.07897659069863168554349251843, 5.21779936128027235843323846850, 6.15702820921683409833157350489, 6.42615960567384731228962631933, 7.83786496577492000887390171228, 8.856129152996388147958189725449, 9.425501114401974431851632864824, 10.56500202005374598910043433634
