| L(s) = 1 | + (−1.16 − 0.799i)2-s + (0.663 − 1.60i)3-s + (0.722 + 1.86i)4-s + (1.22 − 0.624i)5-s + (−2.05 + 1.33i)6-s + (1.33 − 0.820i)7-s + (0.647 − 2.75i)8-s + (−0.00546 − 0.00546i)9-s + (−1.92 − 0.251i)10-s + (0.0312 − 0.396i)11-s + (3.46 + 0.0802i)12-s + (−1.80 + 0.434i)13-s + (−2.21 − 0.112i)14-s + (−0.187 − 2.37i)15-s + (−2.95 + 2.69i)16-s + (−3.70 − 4.33i)17-s + ⋯ |
| L(s) = 1 | + (−0.824 − 0.565i)2-s + (0.383 − 0.925i)3-s + (0.361 + 0.932i)4-s + (0.548 − 0.279i)5-s + (−0.838 + 0.546i)6-s + (0.505 − 0.310i)7-s + (0.229 − 0.973i)8-s + (−0.00182 − 0.00182i)9-s + (−0.610 − 0.0793i)10-s + (0.00942 − 0.119i)11-s + (1.00 + 0.0231i)12-s + (−0.501 + 0.120i)13-s + (−0.592 − 0.0301i)14-s + (−0.0483 − 0.614i)15-s + (−0.739 + 0.673i)16-s + (−0.898 − 1.05i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00799 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 164 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00799 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.692348 - 0.697903i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.692348 - 0.697903i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.16 + 0.799i)T \) |
| 41 | \( 1 + (-0.0405 - 6.40i)T \) |
| good | 3 | \( 1 + (-0.663 + 1.60i)T + (-2.12 - 2.12i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 0.624i)T + (2.93 - 4.04i)T^{2} \) |
| 7 | \( 1 + (-1.33 + 0.820i)T + (3.17 - 6.23i)T^{2} \) |
| 11 | \( 1 + (-0.0312 + 0.396i)T + (-10.8 - 1.72i)T^{2} \) |
| 13 | \( 1 + (1.80 - 0.434i)T + (11.5 - 5.90i)T^{2} \) |
| 17 | \( 1 + (3.70 + 4.33i)T + (-2.65 + 16.7i)T^{2} \) |
| 19 | \( 1 + (0.0591 - 0.246i)T + (-16.9 - 8.62i)T^{2} \) |
| 23 | \( 1 + (-0.527 + 0.383i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.35 + 1.58i)T + (-4.53 - 28.6i)T^{2} \) |
| 31 | \( 1 + (-3.22 - 9.93i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.115 - 0.355i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (2.48 + 0.394i)T + (40.8 + 13.2i)T^{2} \) |
| 47 | \( 1 + (-2.26 - 1.38i)T + (21.3 + 41.8i)T^{2} \) |
| 53 | \( 1 + (-9.34 - 7.98i)T + (8.29 + 52.3i)T^{2} \) |
| 59 | \( 1 + (6.80 + 9.36i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (-5.04 + 0.798i)T + (58.0 - 18.8i)T^{2} \) |
| 67 | \( 1 + (-0.540 + 0.0425i)T + (66.1 - 10.4i)T^{2} \) |
| 71 | \( 1 + (10.9 + 0.862i)T + (70.1 + 11.1i)T^{2} \) |
| 73 | \( 1 + (-3.17 + 3.17i)T - 73iT^{2} \) |
| 79 | \( 1 + (0.650 + 0.269i)T + (55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 - 8.01iT - 83T^{2} \) |
| 89 | \( 1 + (2.54 + 4.15i)T + (-40.4 + 79.2i)T^{2} \) |
| 97 | \( 1 + (18.1 - 1.42i)T + (95.8 - 15.1i)T^{2} \) |
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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
−12.53982064464708490563635403528, −11.61376625520122266919082453012, −10.54321854612425224401102455892, −9.470382156672864757955838064204, −8.504338064599219952131921119998, −7.52923983588965397409096793199, −6.68974432847453908641398568967, −4.70889663024024295437492711747, −2.68512642539412278557489150838, −1.42383563417754264504331486951,
2.24014672915357722912013897419, 4.33948088039289165741202666753, 5.66492479000638123057129527895, 6.83577918928333297074661433676, 8.199223880072748442949978301089, 9.061650363519449514380463306077, 9.984092304980700995293040444168, 10.59144816535915504074138265146, 11.79016991354026834427727752001, 13.36402429529870954300024246634
