L(s) = 1 | + (−2.29 + 0.836i)5-s + (−0.775 − 4.39i)7-s + (−2.73 − 0.996i)11-s + (−2.01 − 1.69i)13-s + (1.67 − 2.89i)17-s + (1.02 + 1.77i)19-s + (1.60 − 9.11i)23-s + (0.754 − 0.632i)25-s + (−5.30 + 4.45i)29-s + (−0.380 + 2.15i)31-s + (5.46 + 9.45i)35-s + (0.708 − 1.22i)37-s + (−3.13 − 2.63i)41-s + (4.42 + 1.61i)43-s + (1.03 + 5.89i)47-s + ⋯ |
L(s) = 1 | + (−1.02 + 0.374i)5-s + (−0.293 − 1.66i)7-s + (−0.825 − 0.300i)11-s + (−0.559 − 0.469i)13-s + (0.405 − 0.701i)17-s + (0.234 + 0.406i)19-s + (0.335 − 1.90i)23-s + (0.150 − 0.126i)25-s + (−0.985 + 0.826i)29-s + (−0.0683 + 0.387i)31-s + (0.923 + 1.59i)35-s + (0.116 − 0.201i)37-s + (−0.489 − 0.410i)41-s + (0.674 + 0.245i)43-s + (0.151 + 0.859i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.505 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.326032 - 0.569073i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.326032 - 0.569073i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (2.29 - 0.836i)T + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.775 + 4.39i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.73 + 0.996i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (2.01 + 1.69i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.67 + 2.89i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.02 - 1.77i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.60 + 9.11i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (5.30 - 4.45i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.380 - 2.15i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.708 + 1.22i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.13 + 2.63i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.42 - 1.61i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.03 - 5.89i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 1.97T + 53T^{2} \) |
| 59 | \( 1 + (-6.20 + 2.25i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.25 + 7.11i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.37 - 1.99i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.60 + 11.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-6.40 - 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.57 + 1.32i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.20 + 1.00i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-6.88 - 11.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (13.9 + 5.07i)T + (74.3 + 62.3i)T^{2} \) |
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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
−10.97609317267952153423980691169, −10.65104818671775099937944857637, −9.636533845376676346748058332200, −8.125943080893700008789364939743, −7.49183115588007251759238671906, −6.76918598465897994527022512560, −5.13053722867253383254968926491, −3.98561607199750286534173226674, −3.01758732012960419152755133305, −0.45395883519180593850241709693,
2.26996924729335978925051992326, 3.64525301439457991439154429527, 5.04250649024083962942455486829, 5.85410522520815102135689532635, 7.35732137451001850425340143688, 8.128466465635901435219070371660, 9.108490927153156578586781507352, 9.870924407671671626150046451295, 11.36652833837483219371062259224, 11.87683912502660570091842885958