| 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
−15.53964079859286507201853967075, −14.61277372705239511553405940791, −12.96426582265224182771363400033, −11.87530765378346682413361224569, −10.76350764612338749088752299927, −9.996156782517383407603986707596, −7.66311581036407141333670781153, −6.03917117156011849340537657314, −3.93543803845657496004371435633, −2.14204777503187763738375889846,
1.32602935166378288414434949806, 4.83460172978971728188011425431, 5.98504728816517354807683366874, 7.41358342141550175791947909899, 8.904373073867965846658813196749, 10.81957825022787953747623735136, 11.97443092908030093048335246392, 13.57182961673363066537789313872, 14.27662338229357277878611130658, 16.12060703288665995525655575398
