| L(s) = 1 | + (6.91 + 13.9i)3-s − 56.7·5-s − 130. i·7-s + (−147. + 193. i)9-s + 484. i·11-s + 555. i·13-s + (−392. − 792. i)15-s + 1.11e3i·17-s + 741.·19-s + (1.82e3 − 900. i)21-s + 459.·23-s + 94.6·25-s + (−3.71e3 − 725. i)27-s + 3.06e3·29-s − 5.98e3i·31-s + ⋯ |
| L(s) = 1 | + (0.443 + 0.896i)3-s − 1.01·5-s − 1.00i·7-s + (−0.606 + 0.794i)9-s + 1.20i·11-s + 0.912i·13-s + (−0.450 − 0.909i)15-s + 0.931i·17-s + 0.470·19-s + (0.900 − 0.445i)21-s + 0.181·23-s + 0.0302·25-s + (−0.981 − 0.191i)27-s + 0.675·29-s − 1.11i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.320 + 0.947i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06966263370\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06966263370\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-6.91 - 13.9i)T \) |
| good | 5 | \( 1 + 56.7T + 3.12e3T^{2} \) |
| 7 | \( 1 + 130. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 484. iT - 1.61e5T^{2} \) |
| 13 | \( 1 - 555. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.11e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 741.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 459.T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.06e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.98e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 3.46e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.49e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 1.68e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.96e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 3.22e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.25e3iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 5.36e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 6.66e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.31e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.03e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.94e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 2.28e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 3.35e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.46e5T + 8.58e9T^{2} \) |
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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
−10.14404407462271719038706149679, −9.510121140579674256771826430527, −8.298222016278001503637180330319, −7.61207302371491145838152263330, −6.63371325205810500564585395478, −4.90015496331831929732288239538, −4.17660388239655530700335670861, −3.48768205269124903887680647971, −1.85353462955962728510747980155, −0.01783718429647416079816023261,
1.10763958612376652431933445146, 2.80638450703650686641319046062, 3.35424236470269758417873969262, 5.10347943733825868357471483857, 6.11354404707292250684726728792, 7.18313808915725158100835322871, 8.216364435176784532540321801242, 8.524412849607252524008261270446, 9.688157683370983088610529531797, 11.18999460213435698695887823450
