L(s) = 1 | + (−1.03 + 0.754i)2-s + (0.598 + 1.84i)3-s + (−0.109 + 0.335i)4-s + (2.14 + 1.55i)5-s + (−2.01 − 1.46i)6-s + (0.230 − 0.708i)7-s + (−0.933 − 2.87i)8-s + (−0.612 + 0.444i)9-s − 3.39·10-s + (−3.17 + 0.969i)11-s − 0.683·12-s + (0.809 − 0.587i)13-s + (0.295 + 0.909i)14-s + (−1.58 + 4.88i)15-s + (2.56 + 1.86i)16-s + (2.09 + 1.52i)17-s + ⋯ |
L(s) = 1 | + (−0.734 + 0.533i)2-s + (0.345 + 1.06i)3-s + (−0.0545 + 0.167i)4-s + (0.957 + 0.695i)5-s + (−0.821 − 0.596i)6-s + (0.0870 − 0.267i)7-s + (−0.329 − 1.01i)8-s + (−0.204 + 0.148i)9-s − 1.07·10-s + (−0.956 + 0.292i)11-s − 0.197·12-s + (0.224 − 0.163i)13-s + (0.0789 + 0.243i)14-s + (−0.409 + 1.26i)15-s + (0.641 + 0.465i)16-s + (0.507 + 0.368i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 143 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.553 - 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.433569 + 0.809162i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.433569 + 0.809162i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + (3.17 - 0.969i)T \) |
| 13 | \( 1 + (-0.809 + 0.587i)T \) |
good | 2 | \( 1 + (1.03 - 0.754i)T + (0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-0.598 - 1.84i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (-2.14 - 1.55i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (-0.230 + 0.708i)T + (-5.66 - 4.11i)T^{2} \) |
| 17 | \( 1 + (-2.09 - 1.52i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (1.34 + 4.14i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 2.54T + 23T^{2} \) |
| 29 | \( 1 + (0.593 - 1.82i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.92 + 3.57i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-2.68 + 8.25i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.11 - 9.57i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 0.689T + 43T^{2} \) |
| 47 | \( 1 + (1.99 + 6.13i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.94 - 4.32i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (0.377 - 1.16i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (-1.52 - 1.11i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 11.2T + 67T^{2} \) |
| 71 | \( 1 + (-11.2 - 8.14i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.90 + 12.0i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.64 - 4.82i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (12.4 + 9.01i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + 6.22T + 89T^{2} \) |
| 97 | \( 1 + (-2.97 + 2.16i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.59066991135648798097610380612, −12.65522968625443388129552088051, −10.87190230213657513879891187603, −10.05999285255476021036066391304, −9.508135899334424174616401534458, −8.352035811823305781689187787229, −7.23651388867622023232491685650, −5.99452679656907369343515555978, −4.35120558849697675800090673385, −2.89651174103091240593398761500,
1.34178406608711536484956939095, 2.44649822960587382305316750625, 5.17544665563195935638546005200, 6.17319302755383943062546707693, 7.88054832360181497480760152636, 8.573113488553209134457416944636, 9.704121518655648674824047259119, 10.47039478019436886145782921680, 11.85685673073489488905808771706, 12.77352207326203285900293314933