| L(s) = 1 | + (9.26 + 6.49i)2-s + (13.1 − 31.8i)3-s + (43.5 + 120. i)4-s + (−379. + 157. i)5-s + (328. − 209. i)6-s + (−712. + 712. i)7-s + (−378. + 1.39e3i)8-s + (707. + 707. i)9-s + (−4.53e3 − 1.01e3i)10-s + (1.05e3 + 2.54e3i)11-s + (4.40e3 + 199. i)12-s + (8.45e3 + 3.50e3i)13-s + (−1.12e4 + 1.97e3i)14-s + 1.41e4i·15-s + (−1.25e4 + 1.04e4i)16-s − 2.17e4i·17-s + ⋯ |
| L(s) = 1 | + (0.818 + 0.574i)2-s + (0.281 − 0.680i)3-s + (0.340 + 0.940i)4-s + (−1.35 + 0.562i)5-s + (0.621 − 0.395i)6-s + (−0.785 + 0.785i)7-s + (−0.261 + 0.965i)8-s + (0.323 + 0.323i)9-s + (−1.43 − 0.319i)10-s + (0.238 + 0.575i)11-s + (0.735 + 0.0332i)12-s + (1.06 + 0.442i)13-s + (−1.09 + 0.192i)14-s + 1.08i·15-s + (−0.768 + 0.640i)16-s − 1.07i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.622 - 0.782i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.817564 + 1.69594i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.817564 + 1.69594i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-9.26 - 6.49i)T \) |
| good | 3 | \( 1 + (-13.1 + 31.8i)T + (-1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (379. - 157. i)T + (5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (712. - 712. i)T - 8.23e5iT^{2} \) |
| 11 | \( 1 + (-1.05e3 - 2.54e3i)T + (-1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-8.45e3 - 3.50e3i)T + (4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 + 2.17e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (4.02e4 + 1.66e4i)T + (6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (1.12e4 + 1.12e4i)T + 3.40e9iT^{2} \) |
| 29 | \( 1 + (7.63e4 - 1.84e5i)T + (-1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 - 1.59e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-3.20e5 + 1.32e5i)T + (6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (-2.37e4 - 2.37e4i)T + 1.94e11iT^{2} \) |
| 43 | \( 1 + (-2.02e5 - 4.89e5i)T + (-1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 + 2.48e4iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-4.59e5 - 1.10e6i)T + (-8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-2.42e6 + 1.00e6i)T + (1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.03e6 - 2.50e6i)T + (-2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (1.39e5 - 3.37e5i)T + (-4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-8.45e5 + 8.45e5i)T - 9.09e12iT^{2} \) |
| 73 | \( 1 + (-3.09e6 - 3.09e6i)T + 1.10e13iT^{2} \) |
| 79 | \( 1 + 3.20e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (8.64e6 + 3.58e6i)T + (1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (-5.70e6 + 5.70e6i)T - 4.42e13iT^{2} \) |
| 97 | \( 1 - 4.67e6T + 8.07e13T^{2} \) |
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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
−15.63248025042946637256834484771, −14.58692456304052265896642433178, −13.19798444914060873107092369653, −12.30479353255743422723309208561, −11.19526604145932101397773929110, −8.700184502179222601617617714847, −7.39051966432835901991739850126, −6.49067720499162064079736111030, −4.27399911088273249924969670325, −2.71486324536279496961164274268,
0.67396945049084301065424776504, 3.75953389283327444196628239031, 4.02647878381795440590023234766, 6.32627462370459106857947379395, 8.359108535753123585903371580236, 10.01886635690521100667303300290, 11.10637549058823969427513000405, 12.45849119565686985370242088657, 13.39416748107724725180116297819, 15.01934358429018490845820017972
