| L(s) = 1 | + (−16.7 + 16.7i)3-s + (−22.1 − 123. i)5-s + (−422. − 422. i)7-s + 169. i·9-s − 1.35e3·11-s + (1.24e3 − 1.24e3i)13-s + (2.42e3 + 1.68e3i)15-s + (−648. − 648. i)17-s + 9.63e3i·19-s + 1.41e4·21-s + (886. − 886. i)23-s + (−1.46e4 + 5.46e3i)25-s + (−1.50e4 − 1.50e4i)27-s − 2.67e4i·29-s − 6.88e3·31-s + ⋯ |
| L(s) = 1 | + (−0.619 + 0.619i)3-s + (−0.177 − 0.984i)5-s + (−1.23 − 1.23i)7-s + 0.232i·9-s − 1.02·11-s + (0.568 − 0.568i)13-s + (0.719 + 0.499i)15-s + (−0.131 − 0.131i)17-s + 1.40i·19-s + 1.52·21-s + (0.0728 − 0.0728i)23-s + (−0.936 + 0.349i)25-s + (−0.763 − 0.763i)27-s − 1.09i·29-s − 0.230·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.743 + 0.668i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.148100 - 0.386369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.148100 - 0.386369i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (22.1 + 123. i)T \) |
| good | 3 | \( 1 + (16.7 - 16.7i)T - 729iT^{2} \) |
| 7 | \( 1 + (422. + 422. i)T + 1.17e5iT^{2} \) |
| 11 | \( 1 + 1.35e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (-1.24e3 + 1.24e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 + (648. + 648. i)T + 2.41e7iT^{2} \) |
| 19 | \( 1 - 9.63e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + (-886. + 886. i)T - 1.48e8iT^{2} \) |
| 29 | \( 1 + 2.67e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 6.88e3T + 8.87e8T^{2} \) |
| 37 | \( 1 + (4.23e4 + 4.23e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 - 1.32e5T + 4.75e9T^{2} \) |
| 43 | \( 1 + (-2.08e4 + 2.08e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + (3.16e4 + 3.16e4i)T + 1.07e10iT^{2} \) |
| 53 | \( 1 + (-8.01e4 + 8.01e4i)T - 2.21e10iT^{2} \) |
| 59 | \( 1 + 2.90e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.53e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + (-1.13e4 - 1.13e4i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 1.79e5T + 1.28e11T^{2} \) |
| 73 | \( 1 + (4.14e5 - 4.14e5i)T - 1.51e11iT^{2} \) |
| 79 | \( 1 + 1.45e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (6.31e4 - 6.31e4i)T - 3.26e11iT^{2} \) |
| 89 | \( 1 + 3.33e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-5.19e5 - 5.19e5i)T + 8.32e11iT^{2} \) |
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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
−16.29865372306267756418706209132, −15.89123710086312055116480365944, −13.57934320140161118879857349755, −12.64833279849642477146546836804, −10.78127475180349642864845892078, −9.831003535769792820642266267023, −7.83826793311872947758391779672, −5.68294929008872680421598238461, −4.00989951124966273499554036944, −0.27452828787882270211525242084,
2.87013618533827115854548418431, 5.93590423495439834479120060935, 6.98485549389751587989967109293, 9.170338664068122482181094191590, 10.92195156438794204391059210241, 12.19132359826948902107552982145, 13.29275611286430748848142474169, 15.15533760388501473674945538843, 16.04112704298128067585001987171, 17.90610113069689817035881000154
