L(s) = 1 | + (1.5 + 0.866i)3-s + (2.88 − 4.98i)5-s − 1.94·7-s + (1.5 + 2.59i)9-s − 8.46·11-s + (−16.7 + 9.66i)13-s + (8.64 − 4.98i)15-s + (12.5 − 21.7i)17-s + (−17.8 − 6.59i)19-s + (−2.91 − 1.68i)21-s + (−15.6 − 27.0i)23-s + (−4.09 − 7.08i)25-s + 5.19i·27-s + (13.7 − 7.91i)29-s − 14.4i·31-s + ⋯ |
L(s) = 1 | + (0.5 + 0.288i)3-s + (0.576 − 0.997i)5-s − 0.277·7-s + (0.166 + 0.288i)9-s − 0.769·11-s + (−1.28 + 0.743i)13-s + (0.576 − 0.332i)15-s + (0.737 − 1.27i)17-s + (−0.937 − 0.347i)19-s + (−0.138 − 0.0801i)21-s + (−0.680 − 1.17i)23-s + (−0.163 − 0.283i)25-s + 0.192i·27-s + (0.472 − 0.272i)29-s − 0.465i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.703 + 0.710i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.109662675\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.109662675\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 - 0.866i)T \) |
| 19 | \( 1 + (17.8 + 6.59i)T \) |
good | 5 | \( 1 + (-2.88 + 4.98i)T + (-12.5 - 21.6i)T^{2} \) |
| 7 | \( 1 + 1.94T + 49T^{2} \) |
| 11 | \( 1 + 8.46T + 121T^{2} \) |
| 13 | \( 1 + (16.7 - 9.66i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (-12.5 + 21.7i)T + (-144.5 - 250. i)T^{2} \) |
| 23 | \( 1 + (15.6 + 27.0i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-13.7 + 7.91i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 14.4iT - 961T^{2} \) |
| 37 | \( 1 + 41.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-1.53 - 0.885i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (14.4 - 25.0i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + (1.70 + 2.95i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (80.0 - 46.2i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (4.27 + 2.46i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-7.45 - 12.9i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-70.4 + 40.6i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + (52.9 + 30.5i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (38.8 - 67.2i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (84.0 + 48.5i)T + (3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-103. + 59.8i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (42.1 + 24.3i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−9.608712938308747656036279764597, −8.893465266223731123068586382150, −8.013626265275127600267178245016, −7.15055977365603238494186334421, −6.01069726673314574862225674915, −4.90332605524740025645185182422, −4.49126151420481762746888035930, −2.88512413899760675762133594860, −2.03102092744390129974719414426, −0.30133813315200498856104728754,
1.79487472944977619141785993823, 2.77376167965285578524015071376, 3.55810158920937104382442766508, 5.05544512933246073031641130515, 6.04701139995043785345587372169, 6.78687829231156662868883857232, 7.78983665192387102625655854464, 8.274940724535651606715008251331, 9.650089799111214576064723364651, 10.18914759596574239518531009326