| L(s) = 1 | + (0.733 + 0.680i)2-s + (−2.17 − 0.328i)3-s + (0.0747 + 0.997i)4-s + (−0.172 − 0.440i)5-s + (−1.37 − 1.72i)6-s + (−0.623 + 0.781i)8-s + (1.77 + 0.546i)9-s + (0.172 − 0.440i)10-s + (0.452 − 0.139i)11-s + (0.164 − 2.19i)12-s + (0.0313 − 0.137i)13-s + (0.231 + 1.01i)15-s + (−0.988 + 0.149i)16-s + (−5.59 − 3.81i)17-s + (0.926 + 1.60i)18-s + (2.32 − 4.02i)19-s + ⋯ |
| L(s) = 1 | + (0.518 + 0.480i)2-s + (−1.25 − 0.189i)3-s + (0.0373 + 0.498i)4-s + (−0.0772 − 0.196i)5-s + (−0.560 − 0.703i)6-s + (−0.220 + 0.276i)8-s + (0.590 + 0.182i)9-s + (0.0546 − 0.139i)10-s + (0.136 − 0.0420i)11-s + (0.0475 − 0.634i)12-s + (0.00869 − 0.0380i)13-s + (0.0598 + 0.262i)15-s + (−0.247 + 0.0372i)16-s + (−1.35 − 0.924i)17-s + (0.218 + 0.378i)18-s + (0.533 − 0.923i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.764654 - 0.458826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.764654 - 0.458826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.733 - 0.680i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (2.17 + 0.328i)T + (2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (0.172 + 0.440i)T + (-3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.452 + 0.139i)T + (9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (-0.0313 + 0.137i)T + (-11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (5.59 + 3.81i)T + (6.21 + 15.8i)T^{2} \) |
| 19 | \( 1 + (-2.32 + 4.02i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.42 + 2.33i)T + (8.40 - 21.4i)T^{2} \) |
| 29 | \( 1 + (4.42 - 2.12i)T + (18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (0.190 + 0.330i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.807 + 10.7i)T + (-36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-6.40 + 8.03i)T + (-9.12 - 39.9i)T^{2} \) |
| 43 | \( 1 + (4.50 + 5.64i)T + (-9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-2.05 - 1.90i)T + (3.51 + 46.8i)T^{2} \) |
| 53 | \( 1 + (0.937 + 12.5i)T + (-52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (4.59 - 11.7i)T + (-43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (0.290 - 3.87i)T + (-60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-4.31 - 7.47i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (2.99 + 1.44i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (-4.65 + 4.32i)T + (5.45 - 72.7i)T^{2} \) |
| 79 | \( 1 + (4.99 - 8.65i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.43 - 10.6i)T + (-74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (2.90 + 0.896i)T + (73.5 + 50.1i)T^{2} \) |
| 97 | \( 1 + 3.36T + 97T^{2} \) |
| 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
−10.81449575556280870846557177918, −9.297850331275556267588843033257, −8.638920660177023414623363156464, −7.10924493483732373922182549396, −6.88648015483758893861523941817, −5.69740607089766168531395216821, −5.06949045577237709084445983636, −4.16867305928970993852625522015, −2.59333601922153807197097906793, −0.48773241578035376109140736764,
1.45035105445617340668893827675, 3.10404792841011138462292981505, 4.31740468481342146021239711306, 5.11036505518120137583282045399, 6.08432037003916981023145568452, 6.65763791724359578466553441395, 7.939169306592309078040956930634, 9.193972898758475767145431169167, 10.06949163724963703037735958253, 11.05858663064962177982492291043
