| L(s) = 1 | + (0.629 − 1.26i)2-s + (2.11 − 0.815i)3-s + (−1.20 − 1.59i)4-s + (0.924 − 1.18i)5-s + (0.297 − 3.19i)6-s + (−0.371 + 1.03i)7-s + (−2.77 + 0.528i)8-s + (1.58 − 1.43i)9-s + (−0.918 − 1.91i)10-s + (−0.264 − 1.17i)11-s + (−3.85 − 2.38i)12-s + (3.20 − 0.885i)13-s + (1.08 + 1.12i)14-s + (0.988 − 3.25i)15-s + (−1.07 + 3.85i)16-s + (−2.73 + 0.829i)17-s + ⋯ |
| L(s) = 1 | + (0.444 − 0.895i)2-s + (1.22 − 0.470i)3-s + (−0.604 − 0.796i)4-s + (0.413 − 0.529i)5-s + (0.121 − 1.30i)6-s + (−0.140 + 0.392i)7-s + (−0.982 + 0.186i)8-s + (0.527 − 0.478i)9-s + (−0.290 − 0.605i)10-s + (−0.0797 − 0.355i)11-s + (−1.11 − 0.688i)12-s + (0.887 − 0.245i)13-s + (0.288 + 0.300i)14-s + (0.255 − 0.841i)15-s + (−0.269 + 0.962i)16-s + (−0.663 + 0.201i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.504 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.24973 - 2.17753i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.24973 - 2.17753i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.629 + 1.26i)T \) |
| good | 3 | \( 1 + (-2.11 + 0.815i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (-0.924 + 1.18i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (0.371 - 1.03i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (0.264 + 1.17i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (-3.20 + 0.885i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (2.73 - 0.829i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (2.24 + 4.45i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (-6.94 + 1.03i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (0.267 + 1.54i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (4.39 - 6.57i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-4.16 + 3.59i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (2.70 - 10.7i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-1.37 - 3.09i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-0.453 - 0.552i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-1.52 + 8.78i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (-0.570 + 2.06i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (-3.94 - 0.0967i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (5.91 - 5.63i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (-0.589 + 0.0289i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (0.135 + 0.378i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (0.670 - 0.0659i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (11.0 + 9.56i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.01 - 0.150i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (2.42 - 0.483i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.80635491060359242183778093740, −9.468268067961130040234720747987, −8.866440288654610232142404260631, −8.429325468275553545082143585608, −6.90667029181424228103208387343, −5.75569483359889492334794770772, −4.69765498518852663710368061581, −3.35382240296309080558519497351, −2.54509350149028459790976722865, −1.33381762668440820830061984891,
2.45037350891596828056150392601, 3.59358910933516914812224731263, 4.29450372124296942612484353057, 5.71223966427745885370624744435, 6.71011170437190092908106678814, 7.53621915708165777490595891996, 8.576933840303189522289672112732, 9.116816559280406368071418604796, 10.05693696214865392311977517713, 11.02526914405963676831009656214
