| L(s) = 1 | + (−0.930 − 0.365i)2-s + (−1.11 − 1.32i)3-s + (0.733 + 0.680i)4-s + (−1.59 − 1.08i)5-s + (0.551 + 1.64i)6-s + (1.91 − 1.82i)7-s + (−0.433 − 0.900i)8-s + (−0.522 + 2.95i)9-s + (1.08 + 1.59i)10-s + (−0.652 − 4.32i)11-s + (0.0868 − 1.72i)12-s + (−2.72 + 2.17i)13-s + (−2.44 + 1.00i)14-s + (0.331 + 3.32i)15-s + (0.0747 + 0.997i)16-s + (−2.73 − 0.844i)17-s + ⋯ |
| L(s) = 1 | + (−0.658 − 0.258i)2-s + (−0.642 − 0.766i)3-s + (0.366 + 0.340i)4-s + (−0.712 − 0.485i)5-s + (0.224 + 0.670i)6-s + (0.722 − 0.691i)7-s + (−0.153 − 0.318i)8-s + (−0.174 + 0.984i)9-s + (0.343 + 0.503i)10-s + (−0.196 − 1.30i)11-s + (0.0250 − 0.499i)12-s + (−0.754 + 0.601i)13-s + (−0.654 + 0.268i)14-s + (0.0855 + 0.857i)15-s + (0.0186 + 0.249i)16-s + (−0.664 − 0.204i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.964 - 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0440082 + 0.329616i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0440082 + 0.329616i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.930 + 0.365i)T \) |
| 3 | \( 1 + (1.11 + 1.32i)T \) |
| 7 | \( 1 + (-1.91 + 1.82i)T \) |
| good | 5 | \( 1 + (1.59 + 1.08i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.652 + 4.32i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (2.72 - 2.17i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (2.73 + 0.844i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (4.83 - 2.78i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.92 - 6.24i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (6.29 - 1.43i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (2.87 + 1.66i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-7.08 + 6.57i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (5.79 - 2.79i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.565 + 0.272i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-0.490 + 1.24i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.51 + 3.78i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (-1.59 + 1.08i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (9.97 + 10.7i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (-2.49 + 4.32i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.32 - 1.89i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-10.7 + 4.20i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-0.441 - 0.764i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-6.56 + 8.23i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-1.87 - 0.282i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 13.3iT - 97T^{2} \) |
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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
−11.22717266912176326543058121068, −10.75561600014442009791719780842, −9.259878489563943349099788561381, −8.130821508975101567142177827895, −7.65971046094274445222461217860, −6.55661019372425477568037785235, −5.21819276616058568122815452895, −3.91821142314640954197582985980, −1.90249264853305692333735418711, −0.31643866884191722180712116265,
2.45249783235360884301749821993, 4.33458283490330924010623720243, 5.19092066816575205479400972998, 6.53576973618549788605509683738, 7.47015205407910772580807762455, 8.563257074911789216718120024699, 9.499195527915024416620857111537, 10.52509968242272440903841991644, 11.10366366089449320569936855883, 11.99452228240811203261039551939
