| L(s) = 1 | + (−0.662 + 1.24i)2-s + (−1.12 − 1.65i)4-s − i·5-s − 1.52·7-s + (2.81 − 0.303i)8-s + (1.24 + 0.662i)10-s − 5.34i·11-s − 0.986i·13-s + (1.01 − 1.90i)14-s + (−1.48 + 3.71i)16-s − 5.84·17-s + 7.86i·19-s + (−1.65 + 1.12i)20-s + (6.68 + 3.54i)22-s − 4.41·23-s + ⋯ |
| L(s) = 1 | + (−0.468 + 0.883i)2-s + (−0.560 − 0.828i)4-s − 0.447i·5-s − 0.577·7-s + (0.994 − 0.107i)8-s + (0.395 + 0.209i)10-s − 1.61i·11-s − 0.273i·13-s + (0.270 − 0.510i)14-s + (−0.371 + 0.928i)16-s − 1.41·17-s + 1.80i·19-s + (−0.370 + 0.250i)20-s + (1.42 + 0.755i)22-s − 0.921·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1666776917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1666776917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.662 - 1.24i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + iT \) |
| good | 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 5.34iT - 11T^{2} \) |
| 13 | \( 1 + 0.986iT - 13T^{2} \) |
| 17 | \( 1 + 5.84T + 17T^{2} \) |
| 19 | \( 1 - 7.86iT - 19T^{2} \) |
| 23 | \( 1 + 4.41T + 23T^{2} \) |
| 29 | \( 1 - 9.95iT - 29T^{2} \) |
| 31 | \( 1 - 3.04T + 31T^{2} \) |
| 37 | \( 1 - 11.2iT - 37T^{2} \) |
| 41 | \( 1 - 8.75T + 41T^{2} \) |
| 43 | \( 1 + 3.01iT - 43T^{2} \) |
| 47 | \( 1 + 9.74T + 47T^{2} \) |
| 53 | \( 1 + 5.83iT - 53T^{2} \) |
| 59 | \( 1 + 5.85iT - 59T^{2} \) |
| 61 | \( 1 - 6.88iT - 61T^{2} \) |
| 67 | \( 1 - 3.97iT - 67T^{2} \) |
| 71 | \( 1 + 10.2T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 + 2.07T + 79T^{2} \) |
| 83 | \( 1 - 4.30iT - 83T^{2} \) |
| 89 | \( 1 + 3.01T + 89T^{2} \) |
| 97 | \( 1 + 5.12T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.12941842984459506647777845798, −9.317985758323770274453964978382, −8.345403690864016891561092039679, −8.220039935726538162984020432997, −6.83628544676283468565251452714, −6.14307963399558011516653128355, −5.48880260916489201675218139042, −4.34650892001050187688064585696, −3.25630062657258478620930300353, −1.43368446043541150154704541053,
0.087183991843352949954285623663, 2.08830525340876734548666976945, 2.68640444635236718433502054415, 4.17208233976460619247567328440, 4.60773990265466329446025363816, 6.27779912270198589721243494981, 7.10355459355521025324465994107, 7.79282888770411481071795176088, 9.022189565990371923328842968853, 9.515663512943277188702691405233
