| L(s) = 1 | + (2.47 − 7.60i)2-s + (−26.1 + 19.0i)3-s + (−51.7 − 37.6i)4-s + (40.3 + 124. i)5-s + (80.0 + 246. i)6-s + (972. + 706. i)7-s + (−414. + 300. i)8-s + (−351. + 1.08e3i)9-s + 1.04e3·10-s + (4.27e3 + 1.10e3i)11-s + 2.07e3·12-s + (−427. + 1.31e3i)13-s + (7.77e3 − 5.65e3i)14-s + (−3.41e3 − 2.48e3i)15-s + (1.26e3 + 3.89e3i)16-s + (4.17e3 + 1.28e4i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.560 + 0.406i)3-s + (−0.404 − 0.293i)4-s + (0.144 + 0.444i)5-s + (0.151 + 0.465i)6-s + (1.07 + 0.778i)7-s + (−0.286 + 0.207i)8-s + (−0.160 + 0.495i)9-s + 0.330·10-s + (0.968 + 0.249i)11-s + 0.346·12-s + (−0.0539 + 0.166i)13-s + (0.757 − 0.550i)14-s + (−0.261 − 0.190i)15-s + (0.0772 + 0.237i)16-s + (0.206 + 0.634i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.741 - 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.39851 + 0.539331i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.39851 + 0.539331i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.47 + 7.60i)T \) |
| 11 | \( 1 + (-4.27e3 - 1.10e3i)T \) |
| good | 3 | \( 1 + (26.1 - 19.0i)T + (675. - 2.07e3i)T^{2} \) |
| 5 | \( 1 + (-40.3 - 124. i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (-972. - 706. i)T + (2.54e5 + 7.83e5i)T^{2} \) |
| 13 | \( 1 + (427. - 1.31e3i)T + (-5.07e7 - 3.68e7i)T^{2} \) |
| 17 | \( 1 + (-4.17e3 - 1.28e4i)T + (-3.31e8 + 2.41e8i)T^{2} \) |
| 19 | \( 1 + (2.33e4 - 1.69e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + 7.38e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + (-8.10e3 - 5.88e3i)T + (5.33e9 + 1.64e10i)T^{2} \) |
| 31 | \( 1 + (9.45e3 - 2.90e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (-4.29e5 - 3.12e5i)T + (2.93e10 + 9.02e10i)T^{2} \) |
| 41 | \( 1 + (-2.01e5 + 1.46e5i)T + (6.01e10 - 1.85e11i)T^{2} \) |
| 43 | \( 1 + 1.18e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-9.77e5 + 7.10e5i)T + (1.56e11 - 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-2.40e5 + 7.40e5i)T + (-9.50e11 - 6.90e11i)T^{2} \) |
| 59 | \( 1 + (1.11e6 + 8.08e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (1.89e5 + 5.83e5i)T + (-2.54e12 + 1.84e12i)T^{2} \) |
| 67 | \( 1 + 2.89e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + (-1.13e6 - 3.48e6i)T + (-7.35e12 + 5.34e12i)T^{2} \) |
| 73 | \( 1 + (-5.76e5 - 4.18e5i)T + (3.41e12 + 1.05e13i)T^{2} \) |
| 79 | \( 1 + (3.42e5 - 1.05e6i)T + (-1.55e13 - 1.12e13i)T^{2} \) |
| 83 | \( 1 + (2.47e6 + 7.61e6i)T + (-2.19e13 + 1.59e13i)T^{2} \) |
| 89 | \( 1 - 3.06e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (5.12e6 - 1.57e7i)T + (-6.53e13 - 4.74e13i)T^{2} \) |
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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
−16.74028881355721872970870493449, −14.99866409887878914126998907343, −14.12350230313319839422217966044, −12.23067281909013034509419682220, −11.29045703335060261321339549874, −10.16226201169948160723679117882, −8.404684115873054771671847945834, −5.94417095300398284681667019953, −4.40258321921233845535116843596, −1.99855380872789294968006667796,
0.897400232771789526194452154272, 4.36985314830412692191155787267, 6.03113242018402548265584927917, 7.49918697505651942620784619516, 9.078402486537179422901376472468, 11.16777066026295045471201709655, 12.39181912083282466573089661303, 13.84854006164077348491673958754, 14.85037423405330319360145800968, 16.58120065223083476678190729494
