| L(s) = 1 | + (11.0 + 11.0i)3-s + (160. + 160. i)5-s − 53.5·7-s + 242. i·9-s + (122. − 122. i)11-s + (−413. + 413. i)13-s + 3.53e3i·15-s − 3.29e3·17-s + (−7.93e3 − 7.93e3i)19-s + (−589. − 589. i)21-s − 1.36e4·23-s + 3.58e4i·25-s + (−2.67e3 + 2.67e3i)27-s + (−5.98e3 + 5.98e3i)29-s − 2.31e4i·31-s + ⋯ |
| L(s) = 1 | + (0.408 + 0.408i)3-s + (1.28 + 1.28i)5-s − 0.156·7-s + 0.333i·9-s + (0.0919 − 0.0919i)11-s + (−0.188 + 0.188i)13-s + 1.04i·15-s − 0.670·17-s + (−1.15 − 1.15i)19-s + (−0.0636 − 0.0636i)21-s − 1.12·23-s + 2.29i·25-s + (−0.136 + 0.136i)27-s + (−0.245 + 0.245i)29-s − 0.778i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.929 + 0.368i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(1.030613582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030613582\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-11.0 - 11.0i)T \) |
| good | 5 | \( 1 + (-160. - 160. i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 53.5T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-122. + 122. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (413. - 413. i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + 3.29e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + (7.93e3 + 7.93e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 1.36e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + (5.98e3 - 5.98e3i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 2.31e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (6.29e4 + 6.29e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 9.07e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (1.01e5 - 1.01e5i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 - 3.26e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (-2.86e4 - 2.86e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (-1.35e5 + 1.35e5i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (-2.47e5 + 2.47e5i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-5.23e4 - 5.23e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 3.81e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 1.24e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 2.23e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-5.68e5 - 5.68e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 8.63e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.32e6T + 8.32e11T^{2} \) |
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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
−10.77411805225439294196290003095, −9.836037940747543388413627843521, −9.327510264365179246658169287998, −8.152640052098573576882949495636, −6.79727895237468519821694135756, −6.33637062609603746129155482522, −5.08684605588130501876541236427, −3.76646334848590250501170454902, −2.58901655621328051659496718216, −1.95013400815908012282381191406,
0.18143624013361790886358116563, 1.59684336861472122344873487805, 2.15741442165751272866175220439, 3.84595560051849736231025036471, 5.05162210159614534623049687367, 5.95165335662209041364110088508, 6.85124872218206966402845233910, 8.393918265654091034787473488958, 8.669778518268691112321850211371, 9.835721510778004000015643620058
