| L(s) = 1 | + (0.0523 + 0.998i)2-s + (2.06 + 1.67i)3-s + (−0.994 + 0.104i)4-s + (−2.14 + 0.645i)5-s + (−1.56 + 2.15i)6-s + (0.156 + 2.64i)7-s + (−0.156 − 0.987i)8-s + (0.847 + 3.98i)9-s + (−0.756 − 2.10i)10-s + (−0.161 − 0.0342i)11-s + (−2.23 − 1.44i)12-s + (−2.20 − 4.32i)13-s + (−2.62 + 0.294i)14-s + (−5.50 − 2.25i)15-s + (0.978 − 0.207i)16-s + (6.17 + 2.37i)17-s + ⋯ |
| L(s) = 1 | + (0.0370 + 0.706i)2-s + (1.19 + 0.966i)3-s + (−0.497 + 0.0522i)4-s + (−0.957 + 0.288i)5-s + (−0.638 + 0.878i)6-s + (0.0591 + 0.998i)7-s + (−0.0553 − 0.349i)8-s + (0.282 + 1.32i)9-s + (−0.239 − 0.665i)10-s + (−0.0485 − 0.0103i)11-s + (−0.644 − 0.418i)12-s + (−0.611 − 1.19i)13-s + (−0.702 + 0.0786i)14-s + (−1.42 − 0.581i)15-s + (0.244 − 0.0519i)16-s + (1.49 + 0.575i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.836 - 0.547i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.443281 + 1.48705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.443281 + 1.48705i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0523 - 0.998i)T \) |
| 5 | \( 1 + (2.14 - 0.645i)T \) |
| 7 | \( 1 + (-0.156 - 2.64i)T \) |
| good | 3 | \( 1 + (-2.06 - 1.67i)T + (0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (0.161 + 0.0342i)T + (10.0 + 4.47i)T^{2} \) |
| 13 | \( 1 + (2.20 + 4.32i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-6.17 - 2.37i)T + (12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (0.151 - 1.44i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (5.38 - 0.282i)T + (22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-4.25 - 5.85i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.99 - 4.48i)T + (-20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-4.73 + 7.29i)T + (-15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (-8.15 - 2.64i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (4.12 + 4.12i)T + 43iT^{2} \) |
| 47 | \( 1 + (-0.247 - 0.645i)T + (-34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (0.924 - 1.14i)T + (-11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-0.841 - 0.934i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (1.85 + 1.66i)T + (6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (2.27 - 5.93i)T + (-49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (6.83 - 4.96i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-11.6 + 7.56i)T + (29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (-6.99 + 15.7i)T + (-52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (2.67 - 0.423i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-4.48 + 4.98i)T + (-9.30 - 88.5i)T^{2} \) |
| 97 | \( 1 + (-0.624 - 0.0988i)T + (92.2 + 29.9i)T^{2} \) |
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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
−12.10804455172755249155089268806, −10.56113520462915162589642127467, −9.898790148760667341621318996059, −8.837912712556711054527439487147, −8.105560864399108899194811279329, −7.60464326534282301741227508401, −5.92551037372211661565818893756, −4.85276352094643290596720718991, −3.65366814477199167496193751842, −2.85836620505187105299420546506,
1.02030846868612104422419507167, 2.52991235884118528809422075640, 3.72179414884735391980323294846, 4.61744662835487917313397723857, 6.65196814178393144190923381843, 7.82020992764650737651005908288, 7.939598533796375761530642718948, 9.291262943149066965771365880455, 10.03079586980556113620022140822, 11.46052030457237458674959203991
