| L(s) = 1 | + (−11.3 − 0.178i)2-s + (−74.6 + 30.9i)3-s + (127. + 4.04i)4-s + (146. − 354. i)5-s + (850. − 336. i)6-s + (990. + 990. i)7-s + (−1.44e3 − 68.5i)8-s + (3.07e3 − 3.07e3i)9-s + (−1.72e3 + 3.98e3i)10-s + (−5.61e3 − 2.32e3i)11-s + (−9.67e3 + 3.65e3i)12-s + (339. + 820. i)13-s + (−1.10e4 − 1.13e4i)14-s + 3.10e4i·15-s + (1.63e4 + 1.03e3i)16-s + 6.01e3i·17-s + ⋯ |
| L(s) = 1 | + (−0.999 − 0.0157i)2-s + (−1.59 + 0.661i)3-s + (0.999 + 0.0315i)4-s + (0.525 − 1.26i)5-s + (1.60 − 0.635i)6-s + (1.09 + 1.09i)7-s + (−0.998 − 0.0473i)8-s + (1.40 − 1.40i)9-s + (−0.545 + 1.26i)10-s + (−1.27 − 0.526i)11-s + (−1.61 + 0.610i)12-s + (0.0428 + 0.103i)13-s + (−1.07 − 1.10i)14-s + 2.37i·15-s + (0.998 + 0.0631i)16-s + 0.296i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.785 - 0.619i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0957121 + 0.275817i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0957121 + 0.275817i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (11.3 + 0.178i)T \) |
| good | 3 | \( 1 + (74.6 - 30.9i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (-146. + 354. i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-990. - 990. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (5.61e3 + 2.32e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-339. - 820. i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 6.01e3iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (-7.75e3 - 1.87e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (1.58e4 - 1.58e4i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.44e5 - 5.98e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.30e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (5.38e4 - 1.29e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (3.94e5 - 3.94e5i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (2.03e5 + 8.40e4i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.15e6iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (-7.94e5 - 3.29e5i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-8.45e5 + 2.04e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (1.31e6 - 5.44e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (2.76e6 - 1.14e6i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (7.52e5 + 7.52e5i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (-1.80e6 + 1.80e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 - 5.10e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (-3.27e6 - 7.90e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (4.47e6 + 4.47e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 - 3.17e6T + 8.07e13T^{2} \) |
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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
−16.22383478924283751446010425766, −15.22377307587469621617948922989, −12.67507934015339302885624098091, −11.68925057082561718217950559021, −10.72734165470690067819909322994, −9.426530079548450516738825681032, −8.180338515553111323228233694043, −5.77097184988364042348517158834, −5.17952895954079393579961805114, −1.47945836378525876394167794474,
0.23429523346727876916786672975, 1.95146033787625917678278284009, 5.43727696789463302978345518938, 6.96404609265328955644988852293, 7.54581950633020999252388952493, 10.34559161665985147954678014127, 10.73895719228581286249284993234, 11.71797650572233258767899550898, 13.41187340818483694541095956361, 15.03011591247846353657729808119
