| L(s) = 1 | + (−0.830 + 1.14i)2-s + (−1.36 − 0.672i)3-s + (−0.620 − 1.90i)4-s + (1.13 − 3.34i)5-s + (1.90 − 1.00i)6-s + (2.22 − 1.42i)7-s + (2.69 + 0.868i)8-s + (−0.417 − 0.543i)9-s + (2.88 + 4.08i)10-s + (−1.88 − 0.123i)11-s + (−0.432 + 3.01i)12-s + (−6.25 − 1.24i)13-s + (−0.215 + 3.73i)14-s + (−3.80 + 3.80i)15-s + (−3.22 + 2.35i)16-s + (−0.836 + 3.12i)17-s + ⋯ |
| L(s) = 1 | + (−0.587 + 0.809i)2-s + (−0.787 − 0.388i)3-s + (−0.310 − 0.950i)4-s + (0.508 − 1.49i)5-s + (0.777 − 0.409i)6-s + (0.841 − 0.539i)7-s + (0.951 + 0.307i)8-s + (−0.139 − 0.181i)9-s + (0.913 + 1.29i)10-s + (−0.569 − 0.0372i)11-s + (−0.124 + 0.869i)12-s + (−1.73 − 0.345i)13-s + (−0.0576 + 0.998i)14-s + (−0.982 + 0.982i)15-s + (−0.807 + 0.589i)16-s + (−0.202 + 0.757i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.575 + 0.818i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.258530 - 0.497813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.258530 - 0.497813i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.830 - 1.14i)T \) |
| 7 | \( 1 + (-2.22 + 1.42i)T \) |
| good | 3 | \( 1 + (1.36 + 0.672i)T + (1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (-1.13 + 3.34i)T + (-3.96 - 3.04i)T^{2} \) |
| 11 | \( 1 + (1.88 + 0.123i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (6.25 + 1.24i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (0.836 - 3.12i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-3.94 - 3.45i)T + (2.47 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-3.49 + 2.67i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (4.54 + 6.79i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (4.58 - 2.64i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.48 + 0.504i)T + (29.3 + 22.5i)T^{2} \) |
| 41 | \( 1 + (-9.60 - 3.97i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (5.74 + 3.84i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-4.47 + 1.19i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (8.06 + 0.528i)T + (52.5 + 6.91i)T^{2} \) |
| 59 | \( 1 + (-3.99 - 4.55i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.377 + 5.75i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (-1.70 - 0.842i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (2.47 + 5.97i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (12.6 - 1.67i)T + (70.5 - 18.8i)T^{2} \) |
| 79 | \( 1 + (1.82 + 6.82i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.334 + 1.68i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (-0.834 + 6.34i)T + (-85.9 - 23.0i)T^{2} \) |
| 97 | \( 1 - 3.69iT - 97T^{2} \) |
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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
−10.60416599456065735474478837352, −9.751387797884687612468181761006, −8.894129545454546383724631355573, −7.922564265082534683211444460344, −7.27071561607261584519472283032, −5.87243365459924085862004835228, −5.28314317520854457603411788567, −4.56739297806827359259393639874, −1.71089689822024214864810561559, −0.46081983866780392891907969001,
2.23585351028878379223264557445, 2.94066333088194084229385682786, 4.79048616803716987364437671179, 5.46990361487493781893322607007, 7.15153573734575738356445267525, 7.53675870301232233578680844180, 9.129789583279880214391846986348, 9.807640732826502633965112667287, 10.71400592776205956153499266454, 11.25635058475279488618545897903
