| L(s) = 1 | + (1.19 − 1.25i)3-s + (0.311 + 0.371i)5-s + (0.958 + 2.63i)7-s + (−0.140 − 2.99i)9-s + (4.24 + 3.56i)11-s + (−0.0238 − 0.135i)13-s + (0.837 + 0.0535i)15-s + (3.57 − 2.06i)17-s + (−4.52 − 2.61i)19-s + (4.44 + 1.94i)21-s + (2.38 + 0.866i)23-s + (0.827 − 4.69i)25-s + (−3.92 − 3.40i)27-s + (−2.79 − 0.493i)29-s + (−2.14 + 5.89i)31-s + ⋯ |
| L(s) = 1 | + (0.690 − 0.723i)3-s + (0.139 + 0.166i)5-s + (0.362 + 0.995i)7-s + (−0.0468 − 0.998i)9-s + (1.27 + 1.07i)11-s + (−0.00662 − 0.0375i)13-s + (0.216 + 0.0138i)15-s + (0.868 − 0.501i)17-s + (−1.03 − 0.599i)19-s + (0.969 + 0.424i)21-s + (0.496 + 0.180i)23-s + (0.165 − 0.938i)25-s + (−0.754 − 0.655i)27-s + (−0.519 − 0.0916i)29-s + (−0.385 + 1.05i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.968 + 0.247i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.88918 - 0.237143i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88918 - 0.237143i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.19 + 1.25i)T \) |
| good | 5 | \( 1 + (-0.311 - 0.371i)T + (-0.868 + 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.958 - 2.63i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (-4.24 - 3.56i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.0238 + 0.135i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-3.57 + 2.06i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (4.52 + 2.61i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.38 - 0.866i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.79 + 0.493i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (2.14 - 5.89i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (4.89 + 8.48i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.97 - 0.877i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (0.705 - 0.840i)T + (-7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.84 + 0.671i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 - 10.9iT - 53T^{2} \) |
| 59 | \( 1 + (-3.92 + 3.29i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (-5.00 + 1.82i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (11.6 - 2.05i)T + (62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-7.77 - 13.4i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.66 - 11.5i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.12 + 0.374i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.26 + 7.15i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (-2.58 - 1.49i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.42 + 3.71i)T + (16.8 + 95.5i)T^{2} \) |
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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
−11.34548081056928691628382993501, −10.00712060152291191123824080839, −9.058579285356753105149330172759, −8.584503384139130419526387257761, −7.29831455038360104874914421298, −6.67828576666558359010016750365, −5.48700635504142542600175452697, −4.10159630977012025114762565150, −2.69526199513554244394947458573, −1.64281986909367609599159910407,
1.53433584724316717581961389626, 3.44879801935859762226708443629, 4.03612253480256263141515724140, 5.29194642652106655349096992426, 6.52091617040216269631305829064, 7.73952868436847328150266103239, 8.550281948991173260847145248666, 9.327326825280876068461436736359, 10.33070516491921669179546072904, 10.95180860563108865564034032220
