| L(s) = 1 | + (−5.65 + 9.79i)2-s + (−40.5 + 23.3i)3-s + (−63.9 − 110. i)4-s + (537. + 310. i)5-s − 529. i·6-s + (−2.28e3 − 734. i)7-s + 1.44e3·8-s + (1.09e3 − 1.89e3i)9-s + (−6.08e3 + 3.51e3i)10-s + (−1.09e4 − 1.89e4i)11-s + (5.18e3 + 2.99e3i)12-s + 5.80e3i·13-s + (2.01e4 − 1.82e4i)14-s − 2.90e4·15-s + (−8.19e3 + 1.41e4i)16-s + (1.32e5 − 7.65e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.5 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (0.860 + 0.496i)5-s − 0.408i·6-s + (−0.952 − 0.306i)7-s + 0.353·8-s + (0.166 − 0.288i)9-s + (−0.608 + 0.351i)10-s + (−0.748 − 1.29i)11-s + (0.249 + 0.144i)12-s + 0.203i·13-s + (0.524 − 0.474i)14-s − 0.573·15-s + (−0.125 + 0.216i)16-s + (1.58 − 0.916i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0857i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.996 - 0.0857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.18335 + 0.0508058i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18335 + 0.0508058i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (5.65 - 9.79i)T \) |
| 3 | \( 1 + (40.5 - 23.3i)T \) |
| 7 | \( 1 + (2.28e3 + 734. i)T \) |
| good | 5 | \( 1 + (-537. - 310. i)T + (1.95e5 + 3.38e5i)T^{2} \) |
| 11 | \( 1 + (1.09e4 + 1.89e4i)T + (-1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 5.80e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.32e5 + 7.65e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.15e5 - 6.65e4i)T + (8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (2.35e5 - 4.07e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 - 1.18e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.33e6 + 7.70e5i)T + (4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-9.64e5 + 1.67e6i)T + (-1.75e12 - 3.04e12i)T^{2} \) |
| 41 | \( 1 + 5.42e4iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 7.30e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (3.34e6 + 1.93e6i)T + (1.19e13 + 2.06e13i)T^{2} \) |
| 53 | \( 1 + (6.89e6 + 1.19e7i)T + (-3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (3.84e6 - 2.21e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-8.07e6 - 4.66e6i)T + (9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-4.06e6 - 7.04e6i)T + (-2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 - 9.55e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.92e6 + 1.11e6i)T + (4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-3.28e7 + 5.68e7i)T + (-7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 5.08e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (1.61e7 + 9.35e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.35e6iT - 7.83e15T^{2} \) |
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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
−14.14370636343441268037961094478, −13.53435989574436334725925185023, −11.74684526545405504452118362159, −10.15533710775249256908005813127, −9.726583714684538698956099789715, −7.82449589337052900213531285207, −6.30155114340848460000812254090, −5.50090679206783180321229601408, −3.18895609326190552509950762328, −0.67064548563739827920274146936,
1.11636898363685007712754766285, 2.69743959085920871869904214076, 4.94754997413821291898411691994, 6.38635161917483215324423286474, 8.097374567378455545592802639515, 9.800355120006242218250267410008, 10.22609823397155296255467263792, 12.27222432279187626552683329024, 12.60820935325948057416924800650, 13.86746638053850427159270997637
