| L(s) = 1 | + (0.111 + 2.99i)3-s + (−2.18 + 2.60i)5-s + (−6.99 − 2.54i)7-s + (−8.97 + 0.667i)9-s + (−4.16 − 4.96i)11-s + (−0.380 + 2.15i)13-s + (−8.04 − 6.25i)15-s + (5.93 + 3.42i)17-s + (−15.2 − 26.3i)19-s + (6.85 − 21.2i)21-s + (6.13 + 16.8i)23-s + (2.33 + 13.2i)25-s + (−2.99 − 26.8i)27-s + (−19.8 + 3.49i)29-s + (−54.7 + 19.9i)31-s + ⋯ |
| L(s) = 1 | + (0.0370 + 0.999i)3-s + (−0.436 + 0.520i)5-s + (−0.999 − 0.363i)7-s + (−0.997 + 0.0741i)9-s + (−0.379 − 0.451i)11-s + (−0.0292 + 0.165i)13-s + (−0.536 − 0.417i)15-s + (0.349 + 0.201i)17-s + (−0.801 − 1.38i)19-s + (0.326 − 1.01i)21-s + (0.266 + 0.732i)23-s + (0.0934 + 0.530i)25-s + (−0.111 − 0.993i)27-s + (−0.683 + 0.120i)29-s + (−1.76 + 0.643i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.941 + 0.337i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0593480 - 0.341648i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0593480 - 0.341648i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.111 - 2.99i)T \) |
| good | 5 | \( 1 + (2.18 - 2.60i)T + (-4.34 - 24.6i)T^{2} \) |
| 7 | \( 1 + (6.99 + 2.54i)T + (37.5 + 31.4i)T^{2} \) |
| 11 | \( 1 + (4.16 + 4.96i)T + (-21.0 + 119. i)T^{2} \) |
| 13 | \( 1 + (0.380 - 2.15i)T + (-158. - 57.8i)T^{2} \) |
| 17 | \( 1 + (-5.93 - 3.42i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (15.2 + 26.3i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-6.13 - 16.8i)T + (-405. + 340. i)T^{2} \) |
| 29 | \( 1 + (19.8 - 3.49i)T + (790. - 287. i)T^{2} \) |
| 31 | \( 1 + (54.7 - 19.9i)T + (736. - 617. i)T^{2} \) |
| 37 | \( 1 + (4.23 - 7.33i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-57.5 - 10.1i)T + (1.57e3 + 574. i)T^{2} \) |
| 43 | \( 1 + (44.8 - 37.6i)T + (321. - 1.82e3i)T^{2} \) |
| 47 | \( 1 + (11.0 - 30.3i)T + (-1.69e3 - 1.41e3i)T^{2} \) |
| 53 | \( 1 - 36.5iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (11.8 - 14.1i)T + (-604. - 3.42e3i)T^{2} \) |
| 61 | \( 1 + (-4.97 - 1.80i)T + (2.85e3 + 2.39e3i)T^{2} \) |
| 67 | \( 1 + (-11.3 + 64.2i)T + (-4.21e3 - 1.53e3i)T^{2} \) |
| 71 | \( 1 + (-59.6 - 34.4i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-19.4 - 33.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-3.14 - 17.8i)T + (-5.86e3 + 2.13e3i)T^{2} \) |
| 83 | \( 1 + (-104. + 18.4i)T + (6.47e3 - 2.35e3i)T^{2} \) |
| 89 | \( 1 + (-38.3 + 22.1i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (56.7 - 47.5i)T + (1.63e3 - 9.26e3i)T^{2} \) |
| 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
−12.70042218109630298237908118088, −11.16250373519350779624782800246, −10.89681550388295215878252918716, −9.677899308930545167965122916876, −8.968559774968430991323354364471, −7.59989099801562923755490674653, −6.48074363319144654411349817912, −5.21403195553615992931874647781, −3.81475335302604044301022866481, −2.98380043925200378106007205003,
0.17915889739621850480563721754, 2.17208221195358226637409175367, 3.69950333349919039522776770137, 5.43200964654220010793489145512, 6.43380079180195537817736537673, 7.54679470869007729157141194508, 8.418307140907555461606312002298, 9.442689092165191324986880828356, 10.64706739857112715341360473252, 11.93127952305797696753817410055
