| L(s) = 1 | + (0.728 + 1.86i)2-s + (−2.33 + 1.88i)3-s + (−2.93 + 2.71i)4-s + (1.28 + 7.31i)5-s + (−5.21 − 2.96i)6-s + (9.38 + 7.87i)7-s + (−7.19 − 3.49i)8-s + (1.88 − 8.80i)9-s + (−12.6 + 7.73i)10-s + (0.485 − 2.75i)11-s + (1.73 − 11.8i)12-s + (−3.34 + 9.19i)13-s + (−7.82 + 23.2i)14-s + (−16.8 − 14.6i)15-s + (1.26 − 15.9i)16-s + (23.4 − 13.5i)17-s + ⋯ |
| L(s) = 1 | + (0.364 + 0.931i)2-s + (−0.777 + 0.628i)3-s + (−0.734 + 0.678i)4-s + (0.257 + 1.46i)5-s + (−0.868 − 0.494i)6-s + (1.34 + 1.12i)7-s + (−0.899 − 0.436i)8-s + (0.209 − 0.977i)9-s + (−1.26 + 0.773i)10-s + (0.0441 − 0.250i)11-s + (0.144 − 0.989i)12-s + (−0.257 + 0.706i)13-s + (−0.559 + 1.65i)14-s + (−1.12 − 0.975i)15-s + (0.0788 − 0.996i)16-s + (1.38 − 0.797i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.952 + 0.303i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.224490 - 1.44402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.224490 - 1.44402i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.728 - 1.86i)T \) |
| 3 | \( 1 + (2.33 - 1.88i)T \) |
| good | 5 | \( 1 + (-1.28 - 7.31i)T + (-23.4 + 8.55i)T^{2} \) |
| 7 | \( 1 + (-9.38 - 7.87i)T + (8.50 + 48.2i)T^{2} \) |
| 11 | \( 1 + (-0.485 + 2.75i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (3.34 - 9.19i)T + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-23.4 + 13.5i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (22.4 + 12.9i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (0.844 + 1.00i)T + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-45.6 + 16.5i)T + (644. - 540. i)T^{2} \) |
| 31 | \( 1 + (31.8 - 26.7i)T + (166. - 946. i)T^{2} \) |
| 37 | \( 1 + (-31.2 + 18.0i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-3.87 + 10.6i)T + (-1.28e3 - 1.08e3i)T^{2} \) |
| 43 | \( 1 + (-24.0 - 4.24i)T + (1.73e3 + 632. i)T^{2} \) |
| 47 | \( 1 + (18.3 - 21.8i)T + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 23.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + (9.31 + 52.8i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (44.6 - 53.1i)T + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (-0.0456 + 0.125i)T + (-3.43e3 - 2.88e3i)T^{2} \) |
| 71 | \( 1 + (-30.7 + 17.7i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (17.4 - 30.1i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (63.5 - 23.1i)T + (4.78e3 - 4.01e3i)T^{2} \) |
| 83 | \( 1 + (98.6 - 35.9i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-24.6 - 14.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-4.44 + 25.2i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
| 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
−12.41182946692946605011904995887, −11.63718870464912598789284869612, −10.85506023216767813809096729036, −9.645570931909666249064810961142, −8.597036064749179495232915095726, −7.30664533243734832234561243671, −6.31777148204632961003705790101, −5.45707476726904425027328689142, −4.44195536250789618272959340552, −2.81360799280584140686789193162,
0.888308597840759864866290602563, 1.68586563305257436516056676161, 4.23349640067665334160172170920, 4.98857590371648843450413719421, 5.89744276497141485920042520419, 7.77597063147338314953764641619, 8.456056494860611748067141209464, 10.08345989712209076945406724393, 10.66844894714074333926777486643, 11.75761712225525491965319926068
