| L(s) = 1 | + (−23.1 + 13.3i)3-s + (16.0 + 27.8i)5-s − 416.·7-s + (−5.85 + 10.1i)9-s + 1.03e3·11-s + (−2.20e3 − 1.27e3i)13-s + (−746. − 431. i)15-s + (3.77e3 + 6.53e3i)17-s + (−1.24e3 − 6.74e3i)19-s + (9.66e3 − 5.58e3i)21-s + (9.77e3 − 1.69e4i)23-s + (7.29e3 − 1.26e4i)25-s − 1.98e4i·27-s + (2.95e4 + 1.70e4i)29-s − 3.71e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.859 + 0.495i)3-s + (0.128 + 0.223i)5-s − 1.21·7-s + (−0.00802 + 0.0139i)9-s + 0.774·11-s + (−1.00 − 0.578i)13-s + (−0.221 − 0.127i)15-s + (0.767 + 1.32i)17-s + (−0.181 − 0.983i)19-s + (1.04 − 0.602i)21-s + (0.803 − 1.39i)23-s + (0.466 − 0.808i)25-s − 1.00i·27-s + (1.21 + 0.700i)29-s − 0.124i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.571 + 0.820i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.671671 - 0.350764i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.671671 - 0.350764i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (1.24e3 + 6.74e3i)T \) |
| good | 3 | \( 1 + (23.1 - 13.3i)T + (364.5 - 631. i)T^{2} \) |
| 5 | \( 1 + (-16.0 - 27.8i)T + (-7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + 416.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.03e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + (2.20e3 + 1.27e3i)T + (2.41e6 + 4.18e6i)T^{2} \) |
| 17 | \( 1 + (-3.77e3 - 6.53e3i)T + (-1.20e7 + 2.09e7i)T^{2} \) |
| 23 | \( 1 + (-9.77e3 + 1.69e4i)T + (-7.40e7 - 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.95e4 - 1.70e4i)T + (2.97e8 + 5.15e8i)T^{2} \) |
| 31 | \( 1 + 3.71e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 8.27e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (7.85e4 - 4.53e4i)T + (2.37e9 - 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-1.06e4 - 1.84e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-4.66e4 + 8.08e4i)T + (-5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 + (-1.01e5 - 5.83e4i)T + (1.10e10 + 1.91e10i)T^{2} \) |
| 59 | \( 1 + (2.06e4 - 1.19e4i)T + (2.10e10 - 3.65e10i)T^{2} \) |
| 61 | \( 1 + (1.58e5 - 2.75e5i)T + (-2.57e10 - 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.04e5 + 1.17e5i)T + (4.52e10 + 7.83e10i)T^{2} \) |
| 71 | \( 1 + (-1.53e5 + 8.88e4i)T + (6.40e10 - 1.10e11i)T^{2} \) |
| 73 | \( 1 + (1.73e5 + 3.00e5i)T + (-7.56e10 + 1.31e11i)T^{2} \) |
| 79 | \( 1 + (2.25e5 - 1.30e5i)T + (1.21e11 - 2.10e11i)T^{2} \) |
| 83 | \( 1 + 6.15e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + (-1.94e4 - 1.12e4i)T + (2.48e11 + 4.30e11i)T^{2} \) |
| 97 | \( 1 + (-8.74e5 + 5.04e5i)T + (4.16e11 - 7.21e11i)T^{2} \) |
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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
−12.84359569053578178991539584808, −12.11903194644838453222872675645, −10.63778683576507891580354901134, −10.11511570531821542983372472902, −8.725509529465824278798359782011, −6.90051902615228704098695637267, −5.94180195850999187058546472911, −4.55092710740188997427340441607, −2.87527044507067200019183114349, −0.38529353738924129363794753347,
1.09090425494615279811938828701, 3.24920286361640889696996590269, 5.16635269787260213519029463167, 6.40654027069857177564171584534, 7.24375433150479280342451026086, 9.191660543620028319472396899722, 9.968350698431141666325329993258, 11.71722823047994353102561523826, 12.12883921421190206018969975085, 13.27327427469630358117137982302
