| L(s) = 1 | + (−0.173 + 0.984i)2-s + (1.53 − 0.797i)3-s + (−0.939 − 0.342i)4-s + (0.258 + 0.710i)5-s + (0.517 + 1.65i)6-s + (0.777 + 1.34i)7-s + (0.5 − 0.866i)8-s + (1.72 − 2.45i)9-s + (−0.744 + 0.131i)10-s + (−0.832 − 0.480i)11-s + (−1.71 + 0.222i)12-s + (0.416 + 0.496i)13-s + (−1.46 + 0.532i)14-s + (0.963 + 0.886i)15-s + (0.766 + 0.642i)16-s + (−6.73 − 1.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.887 − 0.460i)3-s + (−0.469 − 0.171i)4-s + (0.115 + 0.317i)5-s + (0.211 + 0.674i)6-s + (0.294 + 0.509i)7-s + (0.176 − 0.306i)8-s + (0.576 − 0.817i)9-s + (−0.235 + 0.0415i)10-s + (−0.250 − 0.144i)11-s + (−0.495 + 0.0643i)12-s + (0.115 + 0.137i)13-s + (−0.390 + 0.142i)14-s + (0.248 + 0.228i)15-s + (0.191 + 0.160i)16-s + (−1.63 − 0.287i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.16437 + 0.357668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.16437 + 0.357668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 3 | \( 1 + (-1.53 + 0.797i)T \) |
| 19 | \( 1 + (4.14 - 1.35i)T \) |
| good | 5 | \( 1 + (-0.258 - 0.710i)T + (-3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.777 - 1.34i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.832 + 0.480i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.416 - 0.496i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (6.73 + 1.18i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (0.400 - 1.10i)T + (-17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (1.39 + 7.92i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.63 - 1.52i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 4.12iT - 37T^{2} \) |
| 41 | \( 1 + (-4.09 - 3.43i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.34 - 2.67i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-3.11 + 0.548i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (-13.6 - 4.96i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (2.02 - 11.4i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-10.1 - 3.70i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (9.19 - 1.62i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (0.0322 - 0.0117i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (3.04 + 2.55i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (0.893 - 1.06i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-10.4 + 6.05i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (4.68 - 3.92i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (-9.54 - 1.68i)T + (91.1 + 33.1i)T^{2} \) |
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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
−13.71113819160744651827718434956, −13.09017590544745954744973744585, −11.74174975085028860158521199398, −10.30909203884712172871035158265, −8.993632711437307518788746881846, −8.349260455289232811803781732608, −7.11266483986869173538188889105, −6.10948056958198425160789828870, −4.31648620652381081902753913095, −2.38956251176692016356981997063,
2.15103675785412170099445699019, 3.84499743968368701021912315939, 4.93367921925874954131623824208, 7.12265668720309719967157819412, 8.520626612634324272868290105546, 9.125833463492725328150807344063, 10.48505837675190878083447427140, 11.03275780645542203818752664051, 12.76886808034049224538876905992, 13.30305171998286117274556670944
