| 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 | \( 1 + (-9.54 + 1.68i)T + (91.1 - 33.1i)T^{2} \) |
| 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
−13.30305171998286117274556670944, −12.76886808034049224538876905992, −11.03275780645542203818752664051, −10.48505837675190878083447427140, −9.125833463492725328150807344063, −8.520626612634324272868290105546, −7.12265668720309719967157819412, −4.93367921925874954131623824208, −3.84499743968368701021912315939, −2.15103675785412170099445699019,
2.38956251176692016356981997063, 4.31648620652381081902753913095, 6.10948056958198425160789828870, 7.11266483986869173538188889105, 8.349260455289232811803781732608, 8.993632711437307518788746881846, 10.30909203884712172871035158265, 11.74174975085028860158521199398, 13.09017590544745954744973744585, 13.71113819160744651827718434956
