| L(s) = 1 | + (0.819 + 0.573i)2-s + (0.342 + 0.939i)4-s + (−1.53 − 1.62i)5-s + (−0.164 − 0.351i)7-s + (−0.258 + 0.965i)8-s + (−0.323 − 2.21i)10-s + (−3.24 − 3.86i)11-s + (−2.12 − 3.03i)13-s + (0.0674 − 0.382i)14-s + (−0.766 + 0.642i)16-s + (0.0721 + 0.269i)17-s + (−5.80 + 3.35i)19-s + (1.00 − 1.99i)20-s + (−0.439 − 5.02i)22-s + (−4.00 − 1.86i)23-s + ⋯ |
| L(s) = 1 | + (0.579 + 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.685 − 0.727i)5-s + (−0.0620 − 0.133i)7-s + (−0.0915 + 0.341i)8-s + (−0.102 − 0.699i)10-s + (−0.977 − 1.16i)11-s + (−0.589 − 0.841i)13-s + (0.0180 − 0.102i)14-s + (−0.191 + 0.160i)16-s + (0.0175 + 0.0653i)17-s + (−1.33 + 0.769i)19-s + (0.224 − 0.446i)20-s + (−0.0937 − 1.07i)22-s + (−0.836 − 0.389i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.350 + 0.936i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.505776 - 0.729098i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.505776 - 0.729098i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.819 - 0.573i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.53 + 1.62i)T \) |
| good | 7 | \( 1 + (0.164 + 0.351i)T + (-4.49 + 5.36i)T^{2} \) |
| 11 | \( 1 + (3.24 + 3.86i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (2.12 + 3.03i)T + (-4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (-0.0721 - 0.269i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (5.80 - 3.35i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.00 + 1.86i)T + (14.7 + 17.6i)T^{2} \) |
| 29 | \( 1 + (0.711 + 4.03i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-8.64 + 3.14i)T + (23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (-1.02 + 0.275i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (1.25 + 0.221i)T + (38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (-0.614 + 7.02i)T + (-42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (0.552 - 0.257i)T + (30.2 - 36.0i)T^{2} \) |
| 53 | \( 1 + (-7.96 - 7.96i)T + 53iT^{2} \) |
| 59 | \( 1 + (3.03 + 2.54i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (9.72 + 3.54i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (3.80 - 2.66i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (9.02 + 5.20i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (2.01 + 0.539i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.3 + 2.53i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (4.36 - 6.23i)T + (-28.3 - 77.9i)T^{2} \) |
| 89 | \( 1 + (-5.24 - 9.08i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-7.74 - 0.677i)T + (95.5 + 16.8i)T^{2} \) |
| 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
−10.18322254855890680074208682293, −8.792174629412685650757556012695, −8.075675331960703325705163850951, −7.68231957963001348600623599226, −6.26289411976204396545876562855, −5.56568958258798335052142584669, −4.57137958602138734197442233132, −3.71369065772627637684521635988, −2.51642369007732666513844499380, −0.33680152517360316870230988474,
2.11476762246121172892368737613, 2.93326720822444584379790732984, 4.30134774124582306044917369542, 4.79274556473229999692455008854, 6.21183589574132600490421581156, 7.01242780400813885657408522474, 7.74622598790926326752305088679, 8.885347197596928972410807878740, 10.03536641435077004491174647772, 10.48282865138094342836101969573
