| L(s) = 1 | + (−1.01 − 1.20i)2-s + (0.173 + 0.984i)3-s + (−0.0840 + 0.476i)4-s + (0.743 − 0.131i)5-s + (1.01 − 1.20i)6-s + (0.344 + 2.62i)7-s + (−2.06 + 1.19i)8-s + (−0.939 + 0.342i)9-s + (−0.911 − 0.764i)10-s + 1.40·11-s − 0.484·12-s + (1.76 + 1.47i)13-s + (2.81 − 3.07i)14-s + (0.258 + 0.709i)15-s + (4.44 + 1.61i)16-s + (−1.25 + 3.45i)17-s + ⋯ |
| L(s) = 1 | + (−0.716 − 0.853i)2-s + (0.100 + 0.568i)3-s + (−0.0420 + 0.238i)4-s + (0.332 − 0.0586i)5-s + (0.413 − 0.492i)6-s + (0.130 + 0.991i)7-s + (−0.731 + 0.422i)8-s + (−0.313 + 0.114i)9-s + (−0.288 − 0.241i)10-s + 0.423·11-s − 0.139·12-s + (0.488 + 0.409i)13-s + (0.753 − 0.821i)14-s + (0.0666 + 0.183i)15-s + (1.11 + 0.404i)16-s + (−0.304 + 0.836i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 399 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.954 - 0.298i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968339 + 0.147747i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968339 + 0.147747i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.173 - 0.984i)T \) |
| 7 | \( 1 + (-0.344 - 2.62i)T \) |
| 19 | \( 1 + (-2.26 - 3.72i)T \) |
| good | 2 | \( 1 + (1.01 + 1.20i)T + (-0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (-0.743 + 0.131i)T + (4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 - 1.40T + 11T^{2} \) |
| 13 | \( 1 + (-1.76 - 1.47i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (1.25 - 3.45i)T + (-13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (5.33 + 4.47i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (-8.71 - 1.53i)T + (27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-4.74 - 8.22i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.37 + 5.41i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.68 + 4.77i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-0.560 - 0.203i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.24 + 3.40i)T + (-36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 + (9.71 + 1.71i)T + (49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.39 - 1.96i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-1.58 + 1.89i)T + (-10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (0.194 - 0.231i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (4.27 - 11.7i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (6.34 - 1.11i)T + (68.5 - 24.9i)T^{2} \) |
| 79 | \( 1 + (2.92 - 8.04i)T + (-60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (7.69 + 4.44i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.25 + 7.09i)T + (-83.6 - 30.4i)T^{2} \) |
| 97 | \( 1 + (2.01 + 11.4i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.19948436630530088793560750704, −10.25026591576525692532786822025, −9.688976646811426318857363164137, −8.720928004343047619522053217281, −8.283139347164241846295101130854, −6.31099387326323326143644273395, −5.65106586350066816766123744005, −4.18246015009246433667453522228, −2.78657569341147969339220123860, −1.63463907949240287484360435392,
0.867503690445827566232841928057, 2.88222566415455795413507172796, 4.34889795698357218226075840041, 6.03108137494985141098444481955, 6.61513321843475399531776845046, 7.73674757958559321597505191499, 8.036181503551662733131616523461, 9.397171528822793983195020613548, 9.903120106849634836304267040764, 11.31270098264430241740992386548
