L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + 7-s + 0.999·8-s + 2·11-s + (1.5 + 2.59i)13-s + (−0.5 + 0.866i)14-s + (−0.5 + 0.866i)16-s + (2 − 3.46i)17-s + (4 + 1.73i)19-s + (−1 + 1.73i)22-s + (2 + 3.46i)23-s + (2.5 + 4.33i)25-s − 3·26-s + (−0.499 − 0.866i)28-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + 0.377·7-s + 0.353·8-s + 0.603·11-s + (0.416 + 0.720i)13-s + (−0.133 + 0.231i)14-s + (−0.125 + 0.216i)16-s + (0.485 − 0.840i)17-s + (0.917 + 0.397i)19-s + (−0.213 + 0.369i)22-s + (0.417 + 0.722i)23-s + (0.5 + 0.866i)25-s − 0.588·26-s + (−0.0944 − 0.163i)28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.658 - 0.752i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.08854 + 0.493582i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.08854 + 0.493582i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 + (-4 - 1.73i)T \) |
good | 5 | \( 1 + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-2 - 3.46i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + (-2 + 3.46i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.5 + 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-5 - 8.66i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (2 + 3.46i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (7 - 12.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (5.5 + 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-7 + 12.1i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 + (7 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (1 - 1.73i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.54348307164942019598761102173, −10.73515255721021597639373978246, −9.418110536988220635410834132230, −9.034949659383026950616417339883, −7.70689965926038983256584618083, −7.05318714727489721208141843972, −5.84833753542444496259914943911, −4.87586858957521118499595655105, −3.48617559706665704132131698909, −1.44523947780977611155881925702,
1.22373116820263713944100213477, 2.89243811900855163697293673280, 4.10415229931038676578117562284, 5.37755673532302851801213209369, 6.66304521052953388567010465269, 7.88499860384502907608527076965, 8.639031928671348426855592035890, 9.629396660823084285906428724082, 10.58374644622505703347783348294, 11.25716029823693530957412458319