| L(s) = 1 | + (1.51 − 1.30i)2-s + (2.99 + 0.220i)3-s + (0.585 − 3.95i)4-s + (−1.38 + 0.275i)5-s + (4.81 − 3.57i)6-s + (2.34 + 5.65i)7-s + (−4.28 − 6.75i)8-s + (8.90 + 1.31i)9-s + (−1.73 + 2.22i)10-s + (10.1 − 15.1i)11-s + (2.62 − 11.7i)12-s + (−0.311 + 1.56i)13-s + (10.9 + 5.49i)14-s + (−4.20 + 0.519i)15-s + (−15.3 − 4.63i)16-s + (−4.23 + 4.23i)17-s + ⋯ |
| L(s) = 1 | + (0.757 − 0.653i)2-s + (0.997 + 0.0734i)3-s + (0.146 − 0.989i)4-s + (−0.277 + 0.0551i)5-s + (0.803 − 0.595i)6-s + (0.334 + 0.807i)7-s + (−0.535 − 0.844i)8-s + (0.989 + 0.146i)9-s + (−0.173 + 0.222i)10-s + (0.921 − 1.37i)11-s + (0.218 − 0.975i)12-s + (−0.0239 + 0.120i)13-s + (0.780 + 0.392i)14-s + (−0.280 + 0.0346i)15-s + (−0.957 − 0.289i)16-s + (−0.248 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.536 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.62161 - 1.44083i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.62161 - 1.44083i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.51 + 1.30i)T \) |
| 3 | \( 1 + (-2.99 - 0.220i)T \) |
| good | 5 | \( 1 + (1.38 - 0.275i)T + (23.0 - 9.56i)T^{2} \) |
| 7 | \( 1 + (-2.34 - 5.65i)T + (-34.6 + 34.6i)T^{2} \) |
| 11 | \( 1 + (-10.1 + 15.1i)T + (-46.3 - 111. i)T^{2} \) |
| 13 | \( 1 + (0.311 - 1.56i)T + (-156. - 64.6i)T^{2} \) |
| 17 | \( 1 + (4.23 - 4.23i)T - 289iT^{2} \) |
| 19 | \( 1 + (17.6 + 3.52i)T + (333. + 138. i)T^{2} \) |
| 23 | \( 1 + (8.46 - 20.4i)T + (-374. - 374. i)T^{2} \) |
| 29 | \( 1 + (-12.8 - 19.2i)T + (-321. + 776. i)T^{2} \) |
| 31 | \( 1 - 27.8iT - 961T^{2} \) |
| 37 | \( 1 + (24.4 - 4.85i)T + (1.26e3 - 523. i)T^{2} \) |
| 41 | \( 1 + (-2.48 + 6.00i)T + (-1.18e3 - 1.18e3i)T^{2} \) |
| 43 | \( 1 + (19.4 - 29.1i)T + (-707. - 1.70e3i)T^{2} \) |
| 47 | \( 1 + (-30.6 + 30.6i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (-49.8 + 74.5i)T + (-1.07e3 - 2.59e3i)T^{2} \) |
| 59 | \( 1 + (71.1 - 14.1i)T + (3.21e3 - 1.33e3i)T^{2} \) |
| 61 | \( 1 + (-36.0 - 53.9i)T + (-1.42e3 + 3.43e3i)T^{2} \) |
| 67 | \( 1 + (-35.0 - 52.3i)T + (-1.71e3 + 4.14e3i)T^{2} \) |
| 71 | \( 1 + (112. - 46.5i)T + (3.56e3 - 3.56e3i)T^{2} \) |
| 73 | \( 1 + (-50.7 + 122. i)T + (-3.76e3 - 3.76e3i)T^{2} \) |
| 79 | \( 1 + (60.7 - 60.7i)T - 6.24e3iT^{2} \) |
| 83 | \( 1 + (-11.3 + 57.0i)T + (-6.36e3 - 2.63e3i)T^{2} \) |
| 89 | \( 1 + (-4.85 - 11.7i)T + (-5.60e3 + 5.60e3i)T^{2} \) |
| 97 | \( 1 + 17.3iT - 9.40e3T^{2} \) |
| 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
−12.12000897220098095652435133896, −11.38898819341314487578755417338, −10.30387150540313180218271612658, −9.048201383297880759083567642579, −8.471757175262628555055522167093, −6.81793421651666329363062833899, −5.58275042733445535162318340872, −4.11355677446832094641614773737, −3.16484575006693293066218702877, −1.73087718077067848592859962599,
2.21854963224222974178484879464, 4.03047388479457912596764820201, 4.43458809276609876451622554277, 6.44717690431179831104351387341, 7.35402296447591721864104316484, 8.111740153189411806518509919657, 9.229498738930112122333684889843, 10.42954554953481548848415035335, 11.92815707536815235374295962585, 12.63119767518521530495230272137
