| L(s) = 1 | + (1.81 − 2.16i)2-s + (−5.55 − 15.2i)3-s + (−1.38 − 7.87i)4-s + (−3.70 + 21.0i)5-s + (−43.1 − 15.7i)6-s + (−2.05 − 3.56i)7-s + (−19.5 − 11.3i)8-s + (−139. + 117. i)9-s + (38.8 + 46.2i)10-s + (95.4 − 165. i)11-s + (−112. + 64.9i)12-s + (73.9 − 203. i)13-s + (−11.4 − 2.01i)14-s + (341. − 60.2i)15-s + (−60.1 + 21.8i)16-s + (−195. − 164. i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.617 − 1.69i)3-s + (−0.0868 − 0.492i)4-s + (−0.148 + 0.841i)5-s + (−1.19 − 0.436i)6-s + (−0.0419 − 0.0726i)7-s + (−0.306 − 0.176i)8-s + (−1.72 + 1.44i)9-s + (0.388 + 0.462i)10-s + (0.788 − 1.36i)11-s + (−0.781 + 0.451i)12-s + (0.437 − 1.20i)13-s + (−0.0584 − 0.0103i)14-s + (1.51 − 0.267i)15-s + (−0.234 + 0.0855i)16-s + (−0.676 − 0.567i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.218295 - 1.28396i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.218295 - 1.28396i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.81 + 2.16i)T \) |
| 19 | \( 1 + (-119. - 340. i)T \) |
| good | 3 | \( 1 + (5.55 + 15.2i)T + (-62.0 + 52.0i)T^{2} \) |
| 5 | \( 1 + (3.70 - 21.0i)T + (-587. - 213. i)T^{2} \) |
| 7 | \( 1 + (2.05 + 3.56i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-95.4 + 165. i)T + (-7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-73.9 + 203. i)T + (-2.18e4 - 1.83e4i)T^{2} \) |
| 17 | \( 1 + (195. + 164. i)T + (1.45e4 + 8.22e4i)T^{2} \) |
| 23 | \( 1 + (65.4 + 371. i)T + (-2.62e5 + 9.57e4i)T^{2} \) |
| 29 | \( 1 + (-744. - 887. i)T + (-1.22e5 + 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-628. + 362. i)T + (4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.11e3iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-871. - 2.39e3i)T + (-2.16e6 + 1.81e6i)T^{2} \) |
| 43 | \( 1 + (-141. + 803. i)T + (-3.21e6 - 1.16e6i)T^{2} \) |
| 47 | \( 1 + (617. - 518. i)T + (8.47e5 - 4.80e6i)T^{2} \) |
| 53 | \( 1 + (2.33e3 - 412. i)T + (7.41e6 - 2.69e6i)T^{2} \) |
| 59 | \( 1 + (845. - 1.00e3i)T + (-2.10e6 - 1.19e7i)T^{2} \) |
| 61 | \( 1 + (12.2 + 69.4i)T + (-1.30e7 + 4.73e6i)T^{2} \) |
| 67 | \( 1 + (-5.73e3 - 6.83e3i)T + (-3.49e6 + 1.98e7i)T^{2} \) |
| 71 | \( 1 + (669. + 117. i)T + (2.38e7 + 8.69e6i)T^{2} \) |
| 73 | \( 1 + (-593. + 215. i)T + (2.17e7 - 1.82e7i)T^{2} \) |
| 79 | \( 1 + (2.87e3 + 7.91e3i)T + (-2.98e7 + 2.50e7i)T^{2} \) |
| 83 | \( 1 + (-413. - 716. i)T + (-2.37e7 + 4.11e7i)T^{2} \) |
| 89 | \( 1 + (3.85e3 - 1.06e4i)T + (-4.80e7 - 4.03e7i)T^{2} \) |
| 97 | \( 1 + (931. - 1.11e3i)T + (-1.53e7 - 8.71e7i)T^{2} \) |
| 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
−14.45973713207038645566299521812, −13.63374138249076593195750227303, −12.59274880278065339559340793380, −11.50175511505054794481806388518, −10.72860242126231686671846157171, −8.296390123414984428590268908518, −6.77340859151748009063719926369, −5.82469136730440415990970833982, −2.97470702209108729696555665860, −0.889351340105800559214333324638,
4.17223989440367048129548418417, 4.80756696092284809361339089512, 6.48647217242289620159925725753, 8.832296948965129126024302122016, 9.675832352387500609403655217939, 11.33240432217042744346169553390, 12.30266841166253917818932599041, 14.04095438955824670589910047658, 15.37574598047621411173919494933, 15.82628085158045204405402831146
