| L(s) = 1 | + (23.0 − 13.3i)3-s + (68.5 + 39.5i)5-s + (−337. + 62.3i)7-s + (−9.46 + 16.3i)9-s + (−854. − 1.47e3i)11-s − 3.12e3i·13-s + 2.10e3·15-s + (3.52e3 − 2.03e3i)17-s + (−5.08e3 − 2.93e3i)19-s + (−6.95e3 + 5.93e3i)21-s + (6.66e3 − 1.15e4i)23-s + (−4.67e3 − 8.10e3i)25-s + 1.99e4i·27-s + 6.51e3·29-s + (1.03e4 − 5.99e3i)31-s + ⋯ |
| L(s) = 1 | + (0.854 − 0.493i)3-s + (0.548 + 0.316i)5-s + (−0.983 + 0.181i)7-s + (−0.0129 + 0.0224i)9-s + (−0.641 − 1.11i)11-s − 1.42i·13-s + 0.625·15-s + (0.718 − 0.414i)17-s + (−0.741 − 0.428i)19-s + (−0.750 + 0.640i)21-s + (0.547 − 0.948i)23-s + (−0.299 − 0.518i)25-s + 1.01i·27-s + 0.266·29-s + (0.348 − 0.201i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.771185484\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.771185484\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (337. - 62.3i)T \) |
| good | 3 | \( 1 + (-23.0 + 13.3i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-68.5 - 39.5i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 11 | \( 1 + (854. + 1.47e3i)T + (-8.85e5 + 1.53e6i)T^{2} \) |
| 13 | \( 1 + 3.12e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 + (-3.52e3 + 2.03e3i)T + (1.20e7 - 2.09e7i)T^{2} \) |
| 19 | \( 1 + (5.08e3 + 2.93e3i)T + (2.35e7 + 4.07e7i)T^{2} \) |
| 23 | \( 1 + (-6.66e3 + 1.15e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 - 6.51e3T + 5.94e8T^{2} \) |
| 31 | \( 1 + (-1.03e4 + 5.99e3i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + (-2.32e3 + 4.01e3i)T + (-1.28e9 - 2.22e9i)T^{2} \) |
| 41 | \( 1 + 1.93e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 9.16e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + (-5.58e4 - 3.22e4i)T + (5.38e9 + 9.33e9i)T^{2} \) |
| 53 | \( 1 + (7.47e4 + 1.29e5i)T + (-1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (5.28e4 - 3.05e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (8.54e4 + 4.93e4i)T + (2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (1.55e5 + 2.70e5i)T + (-4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 - 4.01e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (5.82e5 - 3.36e5i)T + (7.56e10 - 1.31e11i)T^{2} \) |
| 79 | \( 1 + (1.60e5 - 2.77e5i)T + (-1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 - 8.32e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 + (-3.28e5 - 1.89e5i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 - 1.05e6iT - 8.32e11T^{2} \) |
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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
−12.53384648233157699112857343855, −10.84432894596921965529662677368, −10.00926054901400828862033691812, −8.702294360468436277244579191964, −7.897347289652044007150628569621, −6.51474967867912190750957976307, −5.43454254135255849745393840017, −3.15711655263472753712434670872, −2.55983643745444356410719377616, −0.49087339311155242100557894163,
1.81411792489305779309994628234, 3.25478405085986605148147764197, 4.47907937095541515671987117118, 6.07441783587326367378917466750, 7.34461844518095105775780072304, 8.802566347918525457521613770007, 9.603923133059841262866175046345, 10.22040050340151340922823575657, 11.93142185759661481438563575563, 12.97003425906311513817313367301
