L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.5 − 1.65i)3-s + (−0.499 + 0.866i)4-s − 0.792i·5-s + (1.18 − 1.26i)6-s + (0.5 + 2.59i)7-s − 0.999·8-s + (−2.5 + 1.65i)9-s + (0.686 − 0.396i)10-s + (2.18 + 3.78i)11-s + (1.68 + 0.396i)12-s + (3.5 + 0.866i)13-s + (−2 + 1.73i)14-s + (−1.31 + 0.396i)15-s + (−0.5 − 0.866i)16-s + (2.18 − 3.78i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.288 − 0.957i)3-s + (−0.249 + 0.433i)4-s − 0.354i·5-s + (0.484 − 0.515i)6-s + (0.188 + 0.981i)7-s − 0.353·8-s + (−0.833 + 0.552i)9-s + (0.216 − 0.125i)10-s + (0.659 + 1.14i)11-s + (0.486 + 0.114i)12-s + (0.970 + 0.240i)13-s + (−0.534 + 0.462i)14-s + (−0.339 + 0.102i)15-s + (−0.125 − 0.216i)16-s + (0.530 − 0.918i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.730 - 0.682i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.49513 + 0.589542i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.49513 + 0.589542i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 + (0.5 + 1.65i)T \) |
| 7 | \( 1 + (-0.5 - 2.59i)T \) |
| 13 | \( 1 + (-3.5 - 0.866i)T \) |
good | 5 | \( 1 + 0.792iT - 5T^{2} \) |
| 11 | \( 1 + (-2.18 - 3.78i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-2.18 + 3.78i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.18 - 2.05i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.68 + 2.12i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.18 + 1.26i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 6.74T + 31T^{2} \) |
| 37 | \( 1 + (10.1 - 5.84i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (8.18 - 4.72i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-2 + 3.46i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 0.939iT - 47T^{2} \) |
| 53 | \( 1 + 2.22iT - 53T^{2} \) |
| 59 | \( 1 + (5.31 + 3.06i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.5 - 2.59i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.1 + 5.84i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (8.05 - 13.9i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - 9.62T + 79T^{2} \) |
| 83 | \( 1 - 1.58iT - 83T^{2} \) |
| 89 | \( 1 + (9.30 - 5.37i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.372 - 0.644i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.27942484763488437905682960345, −9.873172556753613301319925247579, −8.697779429956535260643388699494, −8.316165445064589632614062515176, −7.00278946418370196944288660259, −6.50645569732145907421714534583, −5.40754270409293217259494689245, −4.64243794126639984351978408862, −2.95152889460607902624652597497, −1.48874576662674745439857809010,
1.03266414855375778945310815257, 3.26030631328793554358354183050, 3.73304620807696545768218450700, 4.85807504380340642625059993047, 5.93268575811222532677011065666, 6.78419804036510481825922326936, 8.389388240099634917255945399007, 9.017208570568371541836826999242, 10.29759764663416304252129859135, 10.69182203383642355192447857645