| L(s) = 1 | + (−5.58 − 0.924i)2-s + (6.74 + 14.0i)3-s + (30.2 + 10.3i)4-s + (−19.2 − 7.96i)5-s + (−24.6 − 84.6i)6-s + (71.2 − 71.2i)7-s + (−159. − 85.5i)8-s + (−151. + 189. i)9-s + (100. + 62.2i)10-s + (−490. − 202. i)11-s + (59.5 + 495. i)12-s + (−107. − 258. i)13-s + (−463. + 331. i)14-s + (−17.8 − 324. i)15-s + (811. + 624. i)16-s + 984.·17-s + ⋯ |
| L(s) = 1 | + (−0.986 − 0.163i)2-s + (0.433 + 0.901i)3-s + (0.946 + 0.322i)4-s + (−0.344 − 0.142i)5-s + (−0.279 − 0.960i)6-s + (0.549 − 0.549i)7-s + (−0.881 − 0.472i)8-s + (−0.625 + 0.780i)9-s + (0.316 + 0.196i)10-s + (−1.22 − 0.505i)11-s + (0.119 + 0.992i)12-s + (−0.176 − 0.424i)13-s + (−0.631 + 0.452i)14-s + (−0.0205 − 0.371i)15-s + (0.792 + 0.610i)16-s + 0.826·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 96 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.175 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.381138 - 0.455041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.381138 - 0.455041i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.58 + 0.924i)T \) |
| 3 | \( 1 + (-6.74 - 14.0i)T \) |
| good | 5 | \( 1 + (19.2 + 7.96i)T + (2.20e3 + 2.20e3i)T^{2} \) |
| 7 | \( 1 + (-71.2 + 71.2i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 + (490. + 202. i)T + (1.13e5 + 1.13e5i)T^{2} \) |
| 13 | \( 1 + (107. + 258. i)T + (-2.62e5 + 2.62e5i)T^{2} \) |
| 17 | \( 1 - 984.T + 1.41e6T^{2} \) |
| 19 | \( 1 + (1.01e3 - 421. i)T + (1.75e6 - 1.75e6i)T^{2} \) |
| 23 | \( 1 + (-590. + 590. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + (2.25e3 + 5.43e3i)T + (-1.45e7 + 1.45e7i)T^{2} \) |
| 31 | \( 1 + 1.52e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-4.93e3 + 1.19e4i)T + (-4.90e7 - 4.90e7i)T^{2} \) |
| 41 | \( 1 + (3.96e3 + 3.96e3i)T + 1.15e8iT^{2} \) |
| 43 | \( 1 + (-6.34e3 + 1.53e4i)T + (-1.03e8 - 1.03e8i)T^{2} \) |
| 47 | \( 1 - 1.65e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + (5.46e3 - 1.31e4i)T + (-2.95e8 - 2.95e8i)T^{2} \) |
| 59 | \( 1 + (-1.31e4 + 3.17e4i)T + (-5.05e8 - 5.05e8i)T^{2} \) |
| 61 | \( 1 + (-5.07e3 + 2.10e3i)T + (5.97e8 - 5.97e8i)T^{2} \) |
| 67 | \( 1 + (1.82e4 + 4.41e4i)T + (-9.54e8 + 9.54e8i)T^{2} \) |
| 71 | \( 1 + (-1.70e4 - 1.70e4i)T + 1.80e9iT^{2} \) |
| 73 | \( 1 + (-1.55e4 + 1.55e4i)T - 2.07e9iT^{2} \) |
| 79 | \( 1 + 2.84e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-3.09e4 - 7.47e4i)T + (-2.78e9 + 2.78e9i)T^{2} \) |
| 89 | \( 1 + (6.64e4 - 6.64e4i)T - 5.58e9iT^{2} \) |
| 97 | \( 1 + 1.33e5T + 8.58e9T^{2} \) |
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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
−12.50774764741824159646839675939, −11.09261176921845125431876950631, −10.52983256957630107659780239878, −9.513299362061730756200383790247, −8.135304516211230047378287302287, −7.76534875225695380423396687176, −5.63437903171038471639071745095, −3.95826955506181389719972087894, −2.47988809452620666975812671464, −0.29573389809409298471642654985,
1.58228749953469652546466188239, 2.83836193608024615142164904023, 5.44573926909108941680259638605, 6.93591805904714221524089278819, 7.81908273460770309141733924250, 8.615216084012753771372460792754, 9.860199446519171917293317795468, 11.21776757725573456059593690107, 12.07540938819705624189117150386, 13.16904882956034116043991859121
