| L(s) = 1 | + (1.48 + 4.56i)2-s + (−7.28 − 5.29i)3-s + (7.22 − 5.24i)4-s + (18.4 − 56.8i)5-s + (13.3 − 41.1i)6-s + (58.2 − 42.3i)7-s + (159. + 115. i)8-s + (25.0 + 77.0i)9-s + 287.·10-s + (64.9 − 396. i)11-s − 80.3·12-s + (167. + 514. i)13-s + (279. + 203. i)14-s + (−435. + 316. i)15-s + (−203. + 626. i)16-s + (466. − 1.43e3i)17-s + ⋯ |
| L(s) = 1 | + (0.262 + 0.807i)2-s + (−0.467 − 0.339i)3-s + (0.225 − 0.163i)4-s + (0.330 − 1.01i)5-s + (0.151 − 0.466i)6-s + (0.449 − 0.326i)7-s + (0.878 + 0.638i)8-s + (0.103 + 0.317i)9-s + 0.908·10-s + (0.161 − 0.986i)11-s − 0.161·12-s + (0.274 + 0.844i)13-s + (0.381 + 0.277i)14-s + (−0.499 + 0.362i)15-s + (−0.198 + 0.611i)16-s + (0.391 − 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.978 + 0.206i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.85914 - 0.194402i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.85914 - 0.194402i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (7.28 + 5.29i)T \) |
| 11 | \( 1 + (-64.9 + 396. i)T \) |
| good | 2 | \( 1 + (-1.48 - 4.56i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (-18.4 + 56.8i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (-58.2 + 42.3i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (-167. - 514. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-466. + 1.43e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (255. + 185. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 2.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + (5.36e3 - 3.89e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.29e3 - 7.07e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (5.21e3 - 3.78e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-966. - 702. i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 2.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (1.09e3 + 795. i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-3.96e3 - 1.22e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-4.48e3 + 3.26e3i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (4.95e3 - 1.52e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 7.17e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.05e4 + 3.25e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (1.26e4 - 9.17e3i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.36e4 + 4.20e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-6.03e3 + 1.85e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.11e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-2.63e4 - 8.09e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−16.12060703288665995525655575398, −14.27662338229357277878611130658, −13.57182961673363066537789313872, −11.97443092908030093048335246392, −10.81957825022787953747623735136, −8.904373073867965846658813196749, −7.41358342141550175791947909899, −5.98504728816517354807683366874, −4.83460172978971728188011425431, −1.32602935166378288414434949806,
2.14204777503187763738375889846, 3.93543803845657496004371435633, 6.03917117156011849340537657314, 7.66311581036407141333670781153, 9.996156782517383407603986707596, 10.76350764612338749088752299927, 11.87530765378346682413361224569, 12.96426582265224182771363400033, 14.61277372705239511553405940791, 15.53964079859286507201853967075
