| L(s) = 1 | + (0.852 − 2.05i)2-s + (3.47 + 2.32i)3-s + (−0.677 − 0.677i)4-s + (−6.00 + 1.19i)5-s + (7.73 − 5.16i)6-s + (3.11 + 0.618i)7-s + (6.25 − 2.59i)8-s + (3.23 + 7.80i)9-s + (−2.65 + 13.3i)10-s + (11.7 + 17.5i)11-s + (−0.780 − 3.92i)12-s + (0.501 − 0.501i)13-s + (3.92 − 5.87i)14-s + (−23.6 − 9.77i)15-s − 18.9i·16-s + ⋯ |
| L(s) = 1 | + (0.426 − 1.02i)2-s + (1.15 + 0.773i)3-s + (−0.169 − 0.169i)4-s + (−1.20 + 0.238i)5-s + (1.28 − 0.861i)6-s + (0.444 + 0.0884i)7-s + (0.782 − 0.324i)8-s + (0.359 + 0.866i)9-s + (−0.265 + 1.33i)10-s + (1.06 + 1.59i)11-s + (−0.0650 − 0.327i)12-s + (0.0385 − 0.0385i)13-s + (0.280 − 0.419i)14-s + (−1.57 − 0.651i)15-s − 1.18i·16-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.997 + 0.0641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.85996 - 0.0917856i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.85996 - 0.0917856i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 \) |
| good | 2 | \( 1 + (-0.852 + 2.05i)T + (-2.82 - 2.82i)T^{2} \) |
| 3 | \( 1 + (-3.47 - 2.32i)T + (3.44 + 8.31i)T^{2} \) |
| 5 | \( 1 + (6.00 - 1.19i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-3.11 - 0.618i)T + (45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-11.7 - 17.5i)T + (-46.3 + 111. i)T^{2} \) |
| 13 | \( 1 + (-0.501 + 0.501i)T - 169iT^{2} \) |
| 19 | \( 1 + (1.02 - 2.46i)T + (-255. - 255. i)T^{2} \) |
| 23 | \( 1 + (-12.3 + 8.25i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (6.40 + 32.1i)T + (-776. + 321. i)T^{2} \) |
| 31 | \( 1 + (18.6 - 27.9i)T + (-367. - 887. i)T^{2} \) |
| 37 | \( 1 + (-11.6 - 7.79i)T + (523. + 1.26e3i)T^{2} \) |
| 41 | \( 1 + (-0.0247 - 0.00491i)T + (1.55e3 + 643. i)T^{2} \) |
| 43 | \( 1 + (14.4 + 34.9i)T + (-1.30e3 + 1.30e3i)T^{2} \) |
| 47 | \( 1 + (-28.9 + 28.9i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (5.02 - 12.1i)T + (-1.98e3 - 1.98e3i)T^{2} \) |
| 59 | \( 1 + (4.12 - 1.70i)T + (2.46e3 - 2.46e3i)T^{2} \) |
| 61 | \( 1 + (4.54 - 22.8i)T + (-3.43e3 - 1.42e3i)T^{2} \) |
| 67 | \( 1 + 49.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 + (96.2 + 64.3i)T + (1.92e3 + 4.65e3i)T^{2} \) |
| 73 | \( 1 + (116. - 23.2i)T + (4.92e3 - 2.03e3i)T^{2} \) |
| 79 | \( 1 + (38.0 + 57.0i)T + (-2.38e3 + 5.76e3i)T^{2} \) |
| 83 | \( 1 + (38.3 + 15.8i)T + (4.87e3 + 4.87e3i)T^{2} \) |
| 89 | \( 1 + (-102. - 102. i)T + 7.92e3iT^{2} \) |
| 97 | \( 1 + (-13.5 - 68.2i)T + (-8.69e3 + 3.60e3i)T^{2} \) |
| 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
−11.77360887474433367240932883481, −10.69016521764360204247224104573, −9.839374604187351666857649320495, −8.895156162623704444057457012040, −7.80641868520253415745852795023, −7.02971692173277559125510477498, −4.59558432537756655867623915018, −4.07467890074765631398176617722, −3.19302575491351944816083524906, −1.90903655336688701503896418557,
1.33471244958126806023604597271, 3.25794331035713013796791591005, 4.32303570787518399263148941025, 5.78946101775900014534328437876, 6.94851944517734888877768881030, 7.68965304066965565691795509544, 8.366371214836407760415978652804, 9.045535642359630658297049091793, 11.02563603829242591874109198594, 11.57018463786525095076230061818
