L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.377 + 1.69i)3-s + (−0.499 + 0.866i)4-s − 0.188i·5-s + (−1.65 + 0.517i)6-s + (−1.93 − 1.80i)7-s − 0.999·8-s + (−2.71 − 1.27i)9-s + (0.163 − 0.0944i)10-s + (−2.99 − 5.18i)11-s + (−1.27 − 1.17i)12-s + (−3.59 + 0.247i)13-s + (0.596 − 2.57i)14-s + (0.319 + 0.0714i)15-s + (−0.5 − 0.866i)16-s + (−2.16 + 3.74i)17-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.218 + 0.975i)3-s + (−0.249 + 0.433i)4-s − 0.0844i·5-s + (−0.674 + 0.211i)6-s + (−0.731 − 0.682i)7-s − 0.353·8-s + (−0.904 − 0.425i)9-s + (0.0517 − 0.0298i)10-s + (−0.902 − 1.56i)11-s + (−0.368 − 0.338i)12-s + (−0.997 + 0.0687i)13-s + (0.159 − 0.688i)14-s + (0.0824 + 0.0184i)15-s + (−0.125 − 0.216i)16-s + (−0.524 + 0.908i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.381 + 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0632752 - 0.0945619i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0632752 - 0.0945619i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.377 - 1.69i)T \) |
| 7 | \( 1 + (1.93 + 1.80i)T \) |
| 13 | \( 1 + (3.59 - 0.247i)T \) |
good | 5 | \( 1 + 0.188iT - 5T^{2} \) |
| 11 | \( 1 + (2.99 + 5.18i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (2.16 - 3.74i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.70 - 4.69i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.12 - 1.22i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.72 - 3.30i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 7.73T + 31T^{2} \) |
| 37 | \( 1 + (-1.21 + 0.699i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (7.23 - 4.17i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.00 - 3.48i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + 3.00iT - 47T^{2} \) |
| 53 | \( 1 - 0.440iT - 53T^{2} \) |
| 59 | \( 1 + (1.61 + 0.929i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.69 + 0.976i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-9.40 + 5.42i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.71 + 2.96i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 7.64T + 73T^{2} \) |
| 79 | \( 1 + 12.8T + 79T^{2} \) |
| 83 | \( 1 + 16.1iT - 83T^{2} \) |
| 89 | \( 1 + (12.3 - 7.14i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (8.15 - 14.1i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.19311217371166543166834236309, −10.45754291273744825433055345212, −9.768737441816338628702539052040, −8.624212659679444743620029511954, −7.978162865686646607912674443239, −6.60172900175538887031648588959, −5.87750177731509229043113558790, −4.91331606511819940397038576040, −3.85988400694241931133095787060, −3.01236034207627767227142662655,
0.05468978384931780914736672589, 2.29535633951401368790524088495, 2.66551507083097376993019567924, 4.64044930384282224380566690479, 5.37282573686251916148503145712, 6.67891021360156143055904947795, 7.20491466792265219280154268008, 8.451919874783210147118203894372, 9.514894027136142138048127073340, 10.21447122518452304090060938407