| L(s) = 1 | + (−1.41 + i)3-s + 4.24i·5-s + i·7-s + (1.00 − 2.82i)9-s − 4.24·11-s + 2·13-s + (−4.24 − 6i)15-s + 7.07i·17-s + 4i·19-s + (−1 − 1.41i)21-s + 1.41·23-s − 12.9·25-s + (1.41 + 5.00i)27-s + 2.82i·29-s − 2i·31-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s + 1.89i·5-s + 0.377i·7-s + (0.333 − 0.942i)9-s − 1.27·11-s + 0.554·13-s + (−1.09 − 1.54i)15-s + 1.71i·17-s + 0.917i·19-s + (−0.218 − 0.308i)21-s + 0.294·23-s − 2.59·25-s + (0.272 + 0.962i)27-s + 0.525i·29-s − 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.816 + 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7084167070\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7084167070\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 4.24iT - 5T^{2} \) |
| 11 | \( 1 + 4.24T + 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 - 1.41T + 23T^{2} \) |
| 29 | \( 1 - 2.82iT - 29T^{2} \) |
| 31 | \( 1 + 2iT - 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 1.41iT - 41T^{2} \) |
| 43 | \( 1 + 8iT - 43T^{2} \) |
| 47 | \( 1 + 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8iT - 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 4iT - 79T^{2} \) |
| 83 | \( 1 + 2.82T + 83T^{2} \) |
| 89 | \( 1 + 9.89iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.33465030775942445232151551564, −9.672687798382022232401381477909, −8.366169713276494307891421758487, −7.58807071726792152381955214717, −6.55005820381799647192524386325, −6.06152157937162713192241166647, −5.29999027388123524105278016598, −3.89125504032192503000708533328, −3.27332294233879976422299577259, −2.05531601502447260782368620014,
0.35514853221192059713518111999, 1.19904223427491792027089515433, 2.63027988427508299343798363093, 4.41258217807252241215001258390, 4.99396460166988449763428087132, 5.52248525900978772563908915673, 6.63066509997019648397392531406, 7.68689727935459557117602830589, 8.118257297035717582315567101261, 9.167757619846295111712476767178
