| L(s) = 1 | + (−6.06 + 14.3i)3-s + 44.6i·5-s − 49i·7-s + (−169. − 174. i)9-s + 190.·11-s + 542.·13-s + (−640. − 270. i)15-s + 377. i·17-s + 214. i·19-s + (703. + 297. i)21-s + 4.57e3·23-s + 1.13e3·25-s + (3.52e3 − 1.37e3i)27-s − 6.95e3i·29-s + 7.80e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.388 + 0.921i)3-s + 0.798i·5-s − 0.377i·7-s + (−0.697 − 0.716i)9-s + 0.474·11-s + 0.889·13-s + (−0.735 − 0.310i)15-s + 0.316i·17-s + 0.136i·19-s + (0.348 + 0.146i)21-s + 1.80·23-s + 0.362·25-s + (0.931 − 0.363i)27-s − 1.53i·29-s + 1.45i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.123 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.876716860\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.876716860\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.06 - 14.3i)T \) |
| 7 | \( 1 + 49iT \) |
| good | 5 | \( 1 - 44.6iT - 3.12e3T^{2} \) |
| 11 | \( 1 - 190.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 542.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 377. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 214. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 4.57e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 7.80e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 5.56e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 3.05e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 6.65e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 5.92e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.70e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.58e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.12e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 8.07e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.92e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.61e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 5.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.76e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 2.08e4T + 8.58e9T^{2} \) |
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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
−10.99364831599696128581580402811, −10.19655385799431805566097268923, −9.272298127948752902493747228574, −8.303252858026717856564911595685, −6.88433698924241173474021281743, −6.19237869691484811672109291418, −4.94291015071763113784999075900, −3.83986472036780041121544908981, −2.95601449546470558186979807553, −1.01033913924955568342744878041,
0.66893579770625067213958713452, 1.54971415959265470662998726390, 3.03035407441806109530672341641, 4.67605756364705358626964351795, 5.61378961293965153651098932854, 6.58437502488206077915053675718, 7.55532817399618160689862135432, 8.708542838328095488365338149929, 9.167115824871261127430246172135, 10.80317125052748907996395063624
