L(s) = 1 | + (1.5 − 2.59i)3-s + 5-s + (−1.5 − 2.59i)7-s + (−3 − 5.19i)9-s + (−1.5 + 2.59i)11-s + (1 + 3.46i)13-s + (1.5 − 2.59i)15-s + (3.5 + 6.06i)17-s + (−0.5 − 0.866i)19-s − 9·21-s + (3.5 − 6.06i)23-s + 25-s − 9·27-s + (2.5 − 4.33i)29-s − 4·31-s + ⋯ |
L(s) = 1 | + (0.866 − 1.49i)3-s + 0.447·5-s + (−0.566 − 0.981i)7-s + (−1 − 1.73i)9-s + (−0.452 + 0.783i)11-s + (0.277 + 0.960i)13-s + (0.387 − 0.670i)15-s + (0.848 + 1.47i)17-s + (−0.114 − 0.198i)19-s − 1.96·21-s + (0.729 − 1.26i)23-s + 0.200·25-s − 1.73·27-s + (0.464 − 0.804i)29-s − 0.718·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 260 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.15265 - 1.13797i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.15265 - 1.13797i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + (-1 - 3.46i)T \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T + (-1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.5 - 6.06i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.5 + 6.06i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.5 + 4.33i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 + (-1.5 + 2.59i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.5 - 6.06i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 - 7.79i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + (2.5 + 4.33i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.5 - 4.33i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.5 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.5 - 2.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8T + 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (3.5 - 6.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-5.5 - 9.52i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.17554128371401982522629540324, −10.73318432269809254452033174415, −9.741117966510711587374384476735, −8.668143891591581185605195776330, −7.70986097863580839143133130907, −6.88566194168091777047892779835, −6.15033916662252742637679164847, −4.15126644541010437766477122150, −2.68229165966164450814091997592, −1.38598263478942632454871572922,
2.82742163210141865177049829200, 3.38456394538072386425508820427, 5.16667074973538874784221686285, 5.68511246731587629557269549968, 7.59608727551051792063104533307, 8.827687307576528227955970364243, 9.245426829227925592022661067875, 10.18152503572189814322031029344, 10.92717939296625100326435103603, 12.19853878755425947420649886630