L(s) = 1 | + (−1.72 − 2.98i)3-s + (−0.5 − 0.866i)5-s + 4.44·7-s + (−4.44 + 7.70i)9-s + 0.918·11-s + (1.76 − 3.05i)13-s + (−1.72 + 2.98i)15-s + (1.94 + 3.37i)17-s + (4.22 − 1.07i)19-s + (−7.67 − 13.2i)21-s + (3.44 − 5.96i)23-s + (−0.499 + 0.866i)25-s + 20.3·27-s + (0.0407 − 0.0705i)29-s + 3.00·31-s + ⋯ |
L(s) = 1 | + (−0.995 − 1.72i)3-s + (−0.223 − 0.387i)5-s + 1.68·7-s + (−1.48 + 2.56i)9-s + 0.276·11-s + (0.489 − 0.847i)13-s + (−0.445 + 0.771i)15-s + (0.471 + 0.817i)17-s + (0.969 − 0.246i)19-s + (−1.67 − 2.89i)21-s + (0.718 − 1.24i)23-s + (−0.0999 + 0.173i)25-s + 3.91·27-s + (0.00756 − 0.0130i)29-s + 0.539·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.490093789\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.490093789\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-4.22 + 1.07i)T \) |
good | 3 | \( 1 + (1.72 + 2.98i)T + (-1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 - 4.44T + 7T^{2} \) |
| 11 | \( 1 - 0.918T + 11T^{2} \) |
| 13 | \( 1 + (-1.76 + 3.05i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.94 - 3.37i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.44 + 5.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.0407 + 0.0705i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 3.00T + 31T^{2} \) |
| 37 | \( 1 - 6.44T + 37T^{2} \) |
| 41 | \( 1 + (-0.224 - 0.388i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.90 + 3.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.40 - 4.17i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.76 + 4.79i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.04 + 1.80i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.20 - 9.02i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.44 - 7.69i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.46 + 2.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-8.07 - 13.9i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.44 + 11.1i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.16T + 83T^{2} \) |
| 89 | \( 1 + (-1.26 + 2.18i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (3.27 + 5.67i)T + (-48.5 + 84.0i)T^{2} \) |
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
−8.710139336237879011178987136678, −8.134222613611434701314198220644, −7.69722959471659468340102327342, −6.84406421764902513580226992252, −5.87275267910213832229300784963, −5.28943070064734187169733556422, −4.48804837410524372347683544696, −2.68851462583963321207405743097, −1.45765420096126472702759311564, −0.896903678867556048676146006358,
1.19902684311727707524980299997, 3.12226543625697350024152140476, 4.02341980433619974835293590638, 4.79423128275420728677456428947, 5.31918911559413269589635799324, 6.20561883685209392942642719434, 7.30079588686125039787756966054, 8.273918201093456107811261505403, 9.274031333468896978621901208125, 9.666879525238190338816108542012