| L(s) = 1 | + (0.766 − 0.642i)2-s + (1.11 + 1.32i)3-s + (0.173 − 0.984i)4-s + (2.37 − 0.866i)5-s + (1.70 + 0.300i)6-s + (0.173 + 0.984i)7-s + (−0.500 − 0.866i)8-s + (−0.520 + 2.95i)9-s + (1.26 − 2.19i)10-s + (−0.286 − 0.104i)11-s + (1.5 − 0.866i)12-s + (−1.43 − 1.20i)13-s + (0.766 + 0.642i)14-s + (3.79 + 2.19i)15-s + (−0.939 − 0.342i)16-s + (−0.592 + 1.02i)17-s + ⋯ |
| L(s) = 1 | + (0.541 − 0.454i)2-s + (0.642 + 0.766i)3-s + (0.0868 − 0.492i)4-s + (1.06 − 0.387i)5-s + (0.696 + 0.122i)6-s + (0.0656 + 0.372i)7-s + (−0.176 − 0.306i)8-s + (−0.173 + 0.984i)9-s + (0.400 − 0.693i)10-s + (−0.0865 − 0.0314i)11-s + (0.433 − 0.249i)12-s + (−0.399 − 0.335i)13-s + (0.204 + 0.171i)14-s + (0.980 + 0.566i)15-s + (−0.234 − 0.0855i)16-s + (−0.143 + 0.248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 + 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.43029 - 0.141548i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.43029 - 0.141548i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 + 0.642i)T \) |
| 3 | \( 1 + (-1.11 - 1.32i)T \) |
| 7 | \( 1 + (-0.173 - 0.984i)T \) |
| good | 5 | \( 1 + (-2.37 + 0.866i)T + (3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (0.286 + 0.104i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (1.43 + 1.20i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.592 - 1.02i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.31 + 4.01i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.215 - 1.22i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (-2.14 + 1.80i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.847 - 4.80i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-5.41 + 9.37i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.69 + 2.26i)T + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-4.04 - 1.47i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.59 - 9.07i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 11.4T + 53T^{2} \) |
| 59 | \( 1 + (5.01 - 1.82i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.96 + 11.1i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (8.10 + 6.79i)T + (11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (6.76 - 11.7i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (0.163 + 0.283i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-9.60 + 8.06i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (1.18 - 0.994i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-1.89 - 3.28i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.65 - 1.33i)T + (74.3 + 62.3i)T^{2} \) |
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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
−11.12667756842601536067720430422, −10.46086098177633734999498325833, −9.443530494761102369891498847504, −9.043005179862745028072128300800, −7.76698057383685053961081668207, −6.19469640954300286742955724894, −5.25446407535737281669262475341, −4.42752731918502751554233371761, −2.98565728353735754904141128032, −1.99239890228082090450145361907,
1.88344327181844075184732071251, 3.00748067489457519661814878869, 4.42246261116115003567997277719, 5.89938642267690145183335393344, 6.55395570912438910827992279360, 7.47967008440385745968026209833, 8.396491260414410935009462412237, 9.488310591261553953475956209024, 10.33118059729164497597941723603, 11.64935972697113534681436723497
