| L(s) = 1 | + (0.304 + 1.38i)2-s + (1.39 + 1.03i)3-s + (−1.81 + 0.841i)4-s + (0.689 + 1.03i)5-s + (−0.999 + 2.23i)6-s + (0.531 − 1.28i)7-s + (−1.71 − 2.24i)8-s + (0.873 + 2.87i)9-s + (−1.21 + 1.26i)10-s + (0.849 + 0.169i)11-s + (−3.39 − 0.699i)12-s + (−3.13 − 2.09i)13-s + (1.93 + 0.342i)14-s + (−0.104 + 2.14i)15-s + (2.58 − 3.05i)16-s + (−2.43 − 2.43i)17-s + ⋯ |
| L(s) = 1 | + (0.215 + 0.976i)2-s + (0.803 + 0.595i)3-s + (−0.907 + 0.420i)4-s + (0.308 + 0.461i)5-s + (−0.408 + 0.912i)6-s + (0.200 − 0.485i)7-s + (−0.606 − 0.795i)8-s + (0.291 + 0.956i)9-s + (−0.384 + 0.400i)10-s + (0.256 + 0.0509i)11-s + (−0.979 − 0.201i)12-s + (−0.869 − 0.581i)13-s + (0.516 + 0.0916i)14-s + (−0.0270 + 0.554i)15-s + (0.645 − 0.763i)16-s + (−0.591 − 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.362 - 0.931i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.881347 + 1.28879i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.881347 + 1.28879i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.304 - 1.38i)T \) |
| 3 | \( 1 + (-1.39 - 1.03i)T \) |
| good | 5 | \( 1 + (-0.689 - 1.03i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (-0.531 + 1.28i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.849 - 0.169i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (3.13 + 2.09i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (2.43 + 2.43i)T + 17iT^{2} \) |
| 19 | \( 1 + (-4.45 - 2.97i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (1.37 + 3.32i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-0.610 - 3.06i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 4.09T + 31T^{2} \) |
| 37 | \( 1 + (3.06 + 4.58i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (11.6 - 4.83i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-7.69 - 1.53i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (-5.07 + 5.07i)T - 47iT^{2} \) |
| 53 | \( 1 + (-0.596 + 2.99i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (-4.43 + 2.96i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (1.56 + 7.86i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (-2.38 + 0.475i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (14.5 + 6.01i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (13.9 - 5.79i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (10.6 - 10.6i)T - 79iT^{2} \) |
| 83 | \( 1 + (0.102 - 0.153i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (-0.0845 - 0.0350i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 - 1.22iT - 97T^{2} \) |
| 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
−13.27353789228276255635941538619, −12.08102052662062933126115805136, −10.44866479244404442756151811426, −9.787021222360948142777226638068, −8.697916825362965269923784545771, −7.69976627745115859068592748977, −6.83365269231903563691627363593, −5.31322744288202781082470670331, −4.25963311468998898963071610995, −2.90544469540902390831188374857,
1.59489146107074201094994067466, 2.83503052576203429618828440889, 4.34135049034308359802501134672, 5.66251776683025650850609528957, 7.20728918362351157188762749638, 8.632224655629669333162635348946, 9.181119018395035518611683591536, 10.12790516918906492965433166179, 11.70638048583988645985864047032, 12.11524120501775262803325642396
