| L(s) = 1 | + (−1.31 + 1.50i)2-s + (−2.45 + 2.01i)3-s + (−0.547 − 3.96i)4-s + (9.14 + 2.77i)5-s + (0.187 − 6.33i)6-s + (0.716 + 3.60i)7-s + (6.69 + 4.38i)8-s + (0.204 − 1.02i)9-s + (−16.1 + 10.1i)10-s + (0.982 + 9.97i)11-s + (9.30 + 8.60i)12-s + (−6.88 − 22.6i)13-s + (−6.37 − 3.65i)14-s + (−27.9 + 11.5i)15-s + (−15.4 + 4.33i)16-s + (−5.22 + 12.6i)17-s + ⋯ |
| L(s) = 1 | + (−0.656 + 0.753i)2-s + (−0.816 + 0.670i)3-s + (−0.136 − 0.990i)4-s + (1.82 + 0.554i)5-s + (0.0311 − 1.05i)6-s + (0.102 + 0.514i)7-s + (0.836 + 0.547i)8-s + (0.0227 − 0.114i)9-s + (−1.61 + 1.01i)10-s + (0.0892 + 0.906i)11-s + (0.775 + 0.717i)12-s + (−0.529 − 1.74i)13-s + (−0.455 − 0.260i)14-s + (−1.86 + 0.772i)15-s + (−0.962 + 0.271i)16-s + (−0.307 + 0.741i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.734 - 0.679i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.345366 + 0.881892i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.345366 + 0.881892i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.31 - 1.50i)T \) |
| good | 3 | \( 1 + (2.45 - 2.01i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (-9.14 - 2.77i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (-0.716 - 3.60i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.982 - 9.97i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (6.88 + 22.6i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (5.22 - 12.6i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (11.3 - 21.2i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-6.68 + 4.46i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (0.792 - 8.04i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-3.89 - 3.89i)T + 961iT^{2} \) |
| 37 | \( 1 + (-17.2 - 32.3i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (11.4 - 7.67i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (16.0 + 13.2i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-26.8 + 64.7i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (3.44 + 35.0i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (16.2 + 4.94i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (5.93 - 4.87i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (41.0 + 50.0i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-118. + 23.6i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-74.7 - 14.8i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-27.7 - 67.0i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (-48.1 - 25.7i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (59.9 - 89.6i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-28.6 + 28.6i)T - 9.40e3iT^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.68656909702891887665308227119, −12.56204577114167012870235748331, −10.74127363831860979145434879833, −10.25285726244898445725080041011, −9.657666013619516423112015399232, −8.234666023117716045476032816561, −6.64435036004069460221968858210, −5.69876314326021658268465518927, −5.10483941506991204823081674822, −2.09905467082438969033874375816,
0.923357852776421631174515436829, 2.26607157435036037272156371418, 4.69012653601044883675018617607, 6.20631036220291016914856682298, 7.08607199493857353943619401861, 9.025382760997971242943187761652, 9.397762040568879465855879135516, 10.77811300628940582514522091570, 11.56912904691077387014258114469, 12.64292051371889302720720449251
