| 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 | \( 1 + 1.22iT - 97T^{2} \) |
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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
−12.11524120501775262803325642396, −11.70638048583988645985864047032, −10.12790516918906492965433166179, −9.181119018395035518611683591536, −8.632224655629669333162635348946, −7.20728918362351157188762749638, −5.66251776683025650850609528957, −4.34135049034308359802501134672, −2.83503052576203429618828440889, −1.59489146107074201094994067466,
2.90544469540902390831188374857, 4.25963311468998898963071610995, 5.31322744288202781082470670331, 6.83365269231903563691627363593, 7.69976627745115859068592748977, 8.697916825362965269923784545771, 9.787021222360948142777226638068, 10.44866479244404442756151811426, 12.08102052662062933126115805136, 13.27353789228276255635941538619
