| L(s) = 1 | + (1.03 + 3.18i)2-s + (7.28 + 5.29i)3-s + (16.8 − 12.2i)4-s + (32.9 − 101. i)5-s + (−9.31 + 28.6i)6-s + (−32.7 + 23.8i)7-s + (143. + 103. i)8-s + (25.0 + 77.0i)9-s + 356.·10-s + (55.2 + 397. i)11-s + 186.·12-s + (−202. − 621. i)13-s + (−109. − 79.8i)14-s + (775. − 563. i)15-s + (22.3 − 68.8i)16-s + (−601. + 1.84e3i)17-s + ⋯ |
| L(s) = 1 | + (0.183 + 0.563i)2-s + (0.467 + 0.339i)3-s + (0.525 − 0.381i)4-s + (0.588 − 1.81i)5-s + (−0.105 + 0.325i)6-s + (−0.252 + 0.183i)7-s + (0.790 + 0.574i)8-s + (0.103 + 0.317i)9-s + 1.12·10-s + (0.137 + 0.990i)11-s + 0.374·12-s + (−0.331 − 1.02i)13-s + (−0.149 − 0.108i)14-s + (0.889 − 0.646i)15-s + (0.0218 − 0.0672i)16-s + (−0.504 + 1.55i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0922i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.995 - 0.0922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.32150 + 0.107319i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.32150 + 0.107319i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-55.2 - 397. i)T \) |
| good | 2 | \( 1 + (-1.03 - 3.18i)T + (-25.8 + 18.8i)T^{2} \) |
| 5 | \( 1 + (-32.9 + 101. i)T + (-2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (32.7 - 23.8i)T + (5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (202. + 621. i)T + (-3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (601. - 1.84e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-394. - 286. i)T + (7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 805.T + 6.43e6T^{2} \) |
| 29 | \( 1 + (2.94e3 - 2.14e3i)T + (6.33e6 - 1.95e7i)T^{2} \) |
| 31 | \( 1 + (457. + 1.40e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-590. + 429. i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (129. + 94.4i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 + 9.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.82e4 - 1.32e4i)T + (7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (9.20e3 + 2.83e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.06e4 - 2.22e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (7.16e3 - 2.20e4i)T + (-6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 2.71e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-1.04e3 + 3.22e3i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-6.02e4 + 4.37e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (1.84e4 + 5.67e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-8.96e3 + 2.75e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 - 8.83e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (7.36e3 + 2.26e4i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−15.64420148650841648301229552967, −14.80879935121927460697991251524, −13.26940016377605829049184193687, −12.41465072188051083310094159986, −10.36030926900591602957257603588, −9.184625171360675281991175363177, −7.87019757969067237164601163572, −5.86787489578877031597332170372, −4.69175164496650706695334540060, −1.72682569713435187254793638851,
2.32799226118943509538663815602, 3.36963177663452825392164746445, 6.55961291160027536529040215485, 7.32288022722291730663407176910, 9.525325310506877056392442138267, 10.90595346999775706282759850491, 11.69648572051256498830994802384, 13.54024154209244020070173156273, 14.07966682283272756116702955713, 15.50529642447818358431826922296
