| L(s) = 1 | + (−0.494 − 1.93i)2-s + (0.211 − 0.173i)3-s + (−3.51 + 1.91i)4-s + (7.43 + 2.25i)5-s + (−0.439 − 0.323i)6-s + (2.23 + 11.2i)7-s + (5.44 + 5.85i)8-s + (−1.74 + 8.75i)9-s + (0.697 − 15.5i)10-s + (−2.14 − 21.7i)11-s + (−0.409 + 1.01i)12-s + (2.88 + 9.49i)13-s + (20.6 − 9.86i)14-s + (1.95 − 0.811i)15-s + (8.66 − 13.4i)16-s + (3.47 − 8.39i)17-s + ⋯ |
| L(s) = 1 | + (−0.247 − 0.969i)2-s + (0.0703 − 0.0577i)3-s + (−0.877 + 0.478i)4-s + (1.48 + 0.451i)5-s + (−0.0733 − 0.0538i)6-s + (0.318 + 1.60i)7-s + (0.680 + 0.732i)8-s + (−0.193 + 0.972i)9-s + (0.0697 − 1.55i)10-s + (−0.194 − 1.97i)11-s + (−0.0341 + 0.0843i)12-s + (0.221 + 0.730i)13-s + (1.47 − 0.704i)14-s + (0.130 − 0.0541i)15-s + (0.541 − 0.840i)16-s + (0.204 − 0.493i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.262i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.262i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.44633 - 0.192937i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44633 - 0.192937i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.494 + 1.93i)T \) |
| good | 3 | \( 1 + (-0.211 + 0.173i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (-7.43 - 2.25i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-2.23 - 11.2i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (2.14 + 21.7i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-2.88 - 9.49i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-3.47 + 8.39i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (1.26 - 2.36i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-0.199 + 0.133i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-2.56 + 26.0i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (16.5 + 16.5i)T + 961iT^{2} \) |
| 37 | \( 1 + (-24.9 - 46.7i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-2.87 + 1.92i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (29.9 + 24.5i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-6.18 + 14.9i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (2.58 + 26.2i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (-51.8 - 15.7i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-64.3 + 52.8i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (45.8 + 55.8i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (78.8 - 15.6i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (24.0 + 4.78i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (14.9 + 36.2i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-27.5 - 14.7i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (27.4 - 41.1i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-118. + 118. i)T - 9.40e3iT^{2} \) |
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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
−13.28399943359776361639805427810, −11.71677370435880425123159799291, −11.11910690469249816496908279080, −9.971765980960752178432418870671, −8.954323255529162512772034855519, −8.247968756229023186535783927103, −6.01900022764171400263407859151, −5.26140631722925199906128376442, −2.90304006265944599480220744197, −2.00384005222950933211197876035,
1.30893995225734814891540967703, 4.21558227496182432678148181251, 5.38629809039055029639631778327, 6.66100829923650858898746118752, 7.55029528927647703786801589831, 9.007319496421494509578623942928, 9.941739091939276613725994278407, 10.44427514354032448989324650168, 12.70811464268477330183305672126, 13.21427671937630320962180625407
