| L(s) = 1 | + (−22.4 − 15.0i)3-s − 122.·5-s − 206.·7-s + (275. + 674. i)9-s + 142.·11-s − 3.49e3i·13-s + (2.75e3 + 1.84e3i)15-s + 3.38e3i·17-s − 755. i·19-s + (4.63e3 + 3.10e3i)21-s − 1.69e4i·23-s − 561.·25-s + (3.96e3 − 1.92e4i)27-s + 4.11e4·29-s + 2.29e4·31-s + ⋯ |
| L(s) = 1 | + (−0.830 − 0.557i)3-s − 0.981·5-s − 0.602·7-s + (0.378 + 0.925i)9-s + 0.107·11-s − 1.59i·13-s + (0.815 + 0.547i)15-s + 0.688i·17-s − 0.110i·19-s + (0.500 + 0.335i)21-s − 1.38i·23-s − 0.0359·25-s + (0.201 − 0.979i)27-s + 1.68·29-s + 0.769·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.830 - 0.557i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3591410017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3591410017\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (22.4 + 15.0i)T \) |
| good | 5 | \( 1 + 122.T + 1.56e4T^{2} \) |
| 7 | \( 1 + 206.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 142.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 3.49e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 3.38e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 755. iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 1.69e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 - 4.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 - 2.29e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 2.68e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 4.04e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + 1.28e5iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 6.06e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + 5.29e3T + 2.21e10T^{2} \) |
| 59 | \( 1 - 9.73e4T + 4.21e10T^{2} \) |
| 61 | \( 1 - 2.13e5iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 3.85e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 - 3.56e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 1.41e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.22e5T + 2.43e11T^{2} \) |
| 83 | \( 1 - 8.04e3T + 3.26e11T^{2} \) |
| 89 | \( 1 + 4.10e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.52e6T + 8.32e11T^{2} \) |
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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
−10.25693448259502822117917515413, −8.540025531670674452377237382652, −7.898796061014215831048215624096, −6.88192322272359590067204439034, −6.06036081770862503792069477421, −4.97261900383991431766069848471, −3.81579865399070435919516897565, −2.56043296900980092392223334353, −0.819634484598732597117388441498, −0.13915291791865138686664786528,
1.18486296577278221393234427374, 3.10689056580321899390976604942, 4.14551111823940627666689581839, 4.86078419124724132974666152213, 6.26882544419784481230897982253, 6.90394769996818885920821388896, 8.072972108702526049985885864711, 9.354645605883514014752886035933, 9.838963193636460875611879656841, 11.11954056553006043457748044997
