| L(s) = 1 | + (0.970 − 1.02i)2-s + (0.440 − 1.67i)3-s + (−0.116 − 1.99i)4-s + (−0.178 − 0.267i)5-s + (−1.29 − 2.07i)6-s + (−0.862 + 2.08i)7-s + (−2.16 − 1.81i)8-s + (−2.61 − 1.47i)9-s + (−0.449 − 0.0758i)10-s + (1.97 + 0.392i)11-s + (−3.39 − 0.684i)12-s + (5.61 + 3.75i)13-s + (1.30 + 2.90i)14-s + (−0.527 + 0.181i)15-s + (−3.97 + 0.464i)16-s + (1.92 + 1.92i)17-s + ⋯ |
| L(s) = 1 | + (0.686 − 0.727i)2-s + (0.254 − 0.967i)3-s + (−0.0581 − 0.998i)4-s + (−0.0800 − 0.119i)5-s + (−0.528 − 0.848i)6-s + (−0.326 + 0.787i)7-s + (−0.766 − 0.642i)8-s + (−0.870 − 0.491i)9-s + (−0.142 − 0.0239i)10-s + (0.594 + 0.118i)11-s + (−0.980 − 0.197i)12-s + (1.55 + 1.04i)13-s + (0.348 + 0.777i)14-s + (−0.136 + 0.0469i)15-s + (−0.993 + 0.116i)16-s + (0.467 + 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.368 + 0.929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.948065 - 1.39516i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.948065 - 1.39516i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.970 + 1.02i)T \) |
| 3 | \( 1 + (-0.440 + 1.67i)T \) |
| good | 5 | \( 1 + (0.178 + 0.267i)T + (-1.91 + 4.61i)T^{2} \) |
| 7 | \( 1 + (0.862 - 2.08i)T + (-4.94 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-1.97 - 0.392i)T + (10.1 + 4.20i)T^{2} \) |
| 13 | \( 1 + (-5.61 - 3.75i)T + (4.97 + 12.0i)T^{2} \) |
| 17 | \( 1 + (-1.92 - 1.92i)T + 17iT^{2} \) |
| 19 | \( 1 + (3.54 + 2.36i)T + (7.27 + 17.5i)T^{2} \) |
| 23 | \( 1 + (2.19 + 5.29i)T + (-16.2 + 16.2i)T^{2} \) |
| 29 | \( 1 + (-1.06 - 5.33i)T + (-26.7 + 11.0i)T^{2} \) |
| 31 | \( 1 - 4.81T + 31T^{2} \) |
| 37 | \( 1 + (-1.42 - 2.13i)T + (-14.1 + 34.1i)T^{2} \) |
| 41 | \( 1 + (7.50 - 3.10i)T + (28.9 - 28.9i)T^{2} \) |
| 43 | \( 1 + (-0.371 - 0.0738i)T + (39.7 + 16.4i)T^{2} \) |
| 47 | \( 1 + (4.36 - 4.36i)T - 47iT^{2} \) |
| 53 | \( 1 + (1.66 - 8.37i)T + (-48.9 - 20.2i)T^{2} \) |
| 59 | \( 1 + (2.84 - 1.90i)T + (22.5 - 54.5i)T^{2} \) |
| 61 | \( 1 + (0.814 + 4.09i)T + (-56.3 + 23.3i)T^{2} \) |
| 67 | \( 1 + (9.80 - 1.95i)T + (61.8 - 25.6i)T^{2} \) |
| 71 | \( 1 + (4.67 + 1.93i)T + (50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (-3.32 + 1.37i)T + (51.6 - 51.6i)T^{2} \) |
| 79 | \( 1 + (0.888 - 0.888i)T - 79iT^{2} \) |
| 83 | \( 1 + (-7.24 + 10.8i)T + (-31.7 - 76.6i)T^{2} \) |
| 89 | \( 1 + (6.70 + 2.77i)T + (62.9 + 62.9i)T^{2} \) |
| 97 | \( 1 + 17.3iT - 97T^{2} \) |
| 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
−12.27653223936079339272868670997, −11.65010349693064771871162263765, −10.55941075161787838979570490830, −9.064055184657540167496257500595, −8.506761004329325384036766834893, −6.48802415820221311927579827645, −6.19877860513240527411461897691, −4.38849941969182332604893069850, −2.96630817118225517063726917346, −1.54792938547718990493107814434,
3.36606648279997486158733950256, 3.95116816798472786534466276599, 5.40847716873663224427304062688, 6.40555466632224628135912987664, 7.80653603035646483531624012872, 8.636722181029605308768625590787, 9.871188910403879118077530390291, 10.90604766112372346651316676381, 11.86204617646674532077574252089, 13.34800882556134724576894956830
