| 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 | \( 1 - 17.3iT - 9.40e3T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.63119767518521530495230272137, −11.92815707536815235374295962585, −10.42954554953481548848415035335, −9.229498738930112122333684889843, −8.111740153189411806518509919657, −7.35402296447591721864104316484, −6.44717690431179831104351387341, −4.43458809276609876451622554277, −4.03047388479457912596764820201, −2.21854963224222974178484879464,
1.73087718077067848592859962599, 3.16484575006693293066218702877, 4.11355677446832094641614773737, 5.58275042733445535162318340872, 6.81793421651666329363062833899, 8.471757175262628555055522167093, 9.048201383297880759083567642579, 10.30387150540313180218271612658, 11.38898819341314487578755417338, 12.12000897220098095652435133896
