| L(s) = 1 | + 22.1·2-s + (−11.9 − 20.6i)3-s + 360.·4-s + (195. + 338. i)5-s + (−263. − 456. i)6-s + (−569. + 986. i)7-s + 5.13e3·8-s + (809. − 1.40e3i)9-s + (4.31e3 + 7.47e3i)10-s − 6.33e3·11-s + (−4.29e3 − 7.44e3i)12-s + (4.81e3 − 8.33e3i)13-s + (−1.25e4 + 2.18e4i)14-s + (4.65e3 − 8.06e3i)15-s + 6.74e4·16-s + (4.72e3 − 8.19e3i)17-s + ⋯ |
| L(s) = 1 | + 1.95·2-s + (−0.254 − 0.441i)3-s + 2.81·4-s + (0.698 + 1.21i)5-s + (−0.497 − 0.862i)6-s + (−0.627 + 1.08i)7-s + 3.54·8-s + (0.370 − 0.640i)9-s + (1.36 + 2.36i)10-s − 1.43·11-s + (−0.717 − 1.24i)12-s + (0.607 − 1.05i)13-s + (−1.22 + 2.12i)14-s + (0.356 − 0.616i)15-s + 4.11·16-s + (0.233 − 0.404i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(5.62000 + 0.656415i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.62000 + 0.656415i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.55e5 - 2.54e5i)T \) |
| good | 2 | \( 1 - 22.1T + 128T^{2} \) |
| 3 | \( 1 + (11.9 + 20.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (-195. - 338. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (569. - 986. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 6.33e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-4.81e3 + 8.33e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-4.72e3 + 8.19e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.50e3 + 6.07e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.13e4 + 3.69e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (1.04e5 - 1.80e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (9.67e4 + 1.67e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (2.83e4 + 4.91e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.05e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 3.67e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (3.58e5 + 6.21e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 3.52e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (2.62e5 - 4.54e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.10e6 - 1.91e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (1.35e6 - 2.34e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (1.12e5 - 1.95e5i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.04e6 - 1.80e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.21e6 + 2.10e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (2.95e6 + 5.11e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.19e7T + 8.07e13T^{2} \) |
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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
−14.43669357429042623739307056906, −13.00781938791746780294482801756, −12.75147818819236678761831876287, −11.27409310802755406979308796909, −10.22853798308313983764802663634, −7.33955784153022778661827865814, −6.16686290593709191819387040986, −5.54309626870568980253805331827, −3.24082777041614797430325183805, −2.36884898953141362655057630549,
1.80508531803367022260005077090, 3.88951750508331057854503183697, 4.91317097854000072887772815985, 5.94041507165868077350638116022, 7.55264345471672540062183315393, 10.04743594245987229834072987069, 11.02880438084236097493474742503, 12.61152774863464700185617723466, 13.36384474466666817259166169084, 13.83682619307031004166263782221
