L(s) = 1 | − 3.58·2-s + (0.133 + 15.5i)3-s − 19.1·4-s + 11.8i·5-s + (−0.480 − 55.9i)6-s − 150. i·7-s + 183.·8-s + (−242. + 4.17i)9-s − 42.6i·10-s + (−352. − 191. i)11-s + (−2.56 − 297. i)12-s − 778. i·13-s + 541. i·14-s + (−185. + 1.59i)15-s − 47.1·16-s − 1.25e3·17-s + ⋯ |
L(s) = 1 | − 0.634·2-s + (0.00859 + 0.999i)3-s − 0.597·4-s + 0.212i·5-s + (−0.00545 − 0.634i)6-s − 1.16i·7-s + 1.01·8-s + (−0.999 + 0.0171i)9-s − 0.134i·10-s + (−0.878 − 0.477i)11-s + (−0.00513 − 0.597i)12-s − 1.27i·13-s + 0.738i·14-s + (−0.212 + 0.00182i)15-s − 0.0460·16-s − 1.05·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.469 + 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.124862 - 0.207903i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.124862 - 0.207903i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.133 - 15.5i)T \) |
| 11 | \( 1 + (352. + 191. i)T \) |
good | 2 | \( 1 + 3.58T + 32T^{2} \) |
| 5 | \( 1 - 11.8iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 150. iT - 1.68e4T^{2} \) |
| 13 | \( 1 + 778. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.25e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.21e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 880. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.20e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.44e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.31e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 5.39e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.08e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.12e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.39e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.29e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 2.72e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.02e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.64e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 - 1.20e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 1.10e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 5.15e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 8.41e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 9.93e4T + 8.58e9T^{2} \) |
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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
−15.48573640427334074422795067618, −14.12595372014904097782333430333, −13.10693538077639394781762899150, −10.69717170682866839326652435687, −10.42683808438370123724284611310, −8.914095763520900304078698336044, −7.66655399637113535137897411094, −5.23869708196363689932294637787, −3.67154108912693610473334433957, −0.17793631367946754520878291401,
2.05634583321655603632335110553, 5.06797202657983422753352010887, 6.95321199731445064351198618126, 8.489776571803492698139998347429, 9.238573725356789214494365014968, 11.17218953004746087871150696415, 12.57337269571818819580876078093, 13.42644442750317682722670044555, 14.79811757943134527985049793441, 16.36291949900243955152896521984