| L(s) = 1 | + (0.642 − 0.766i)2-s + (0.766 + 0.642i)3-s + (−0.173 − 0.984i)4-s + (−1.97 − 0.348i)5-s + (0.984 − 0.173i)6-s + (−1.45 + 2.21i)7-s + (−0.866 − 0.500i)8-s + (0.173 + 0.984i)9-s + (−1.53 + 1.29i)10-s + (1.07 + 1.86i)11-s + (0.500 − 0.866i)12-s + (0.220 + 1.25i)13-s + (0.759 + 2.53i)14-s + (−1.29 − 1.53i)15-s + (−0.939 + 0.342i)16-s + (7.03 + 1.23i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (0.442 + 0.371i)3-s + (−0.0868 − 0.492i)4-s + (−0.885 − 0.156i)5-s + (0.402 − 0.0708i)6-s + (−0.549 + 0.835i)7-s + (−0.306 − 0.176i)8-s + (0.0578 + 0.328i)9-s + (−0.486 + 0.408i)10-s + (0.324 + 0.562i)11-s + (0.144 − 0.249i)12-s + (0.0612 + 0.347i)13-s + (0.202 + 0.677i)14-s + (−0.333 − 0.397i)15-s + (−0.234 + 0.0855i)16-s + (1.70 + 0.300i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 - 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.66435 + 0.564650i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.66435 + 0.564650i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (1.45 - 2.21i)T \) |
| 19 | \( 1 + (-3.16 - 2.99i)T \) |
| good | 5 | \( 1 + (1.97 + 0.348i)T + (4.69 + 1.71i)T^{2} \) |
| 11 | \( 1 + (-1.07 - 1.86i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.220 - 1.25i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-7.03 - 1.23i)T + (15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-6.37 - 2.32i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (2.83 - 7.77i)T + (-22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 - 1.43T + 31T^{2} \) |
| 37 | \( 1 + (1.70 - 0.984i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.207 - 1.17i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (8.46 + 7.10i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (0.168 - 0.0296i)T + (44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (5.47 - 0.965i)T + (49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-0.185 + 1.05i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.526 - 1.44i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (3.88 + 4.62i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (1.29 - 1.53i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-1.19 + 1.42i)T + (-12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.67 - 7.33i)T + (-60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-3.98 + 2.29i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-3.37 + 2.83i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (2.01 - 0.731i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.27142429187603110201916889146, −9.579023047949412981245112140676, −8.855649469610801511251373033445, −7.88015271316700531333179677572, −6.94455032799046760763306458988, −5.64416944526610598242040046246, −4.88415860339894015653820708749, −3.58590783668615106287065918338, −3.22052496706531781766898083526, −1.59638040274557397197830331914,
0.78251013334802608944422778475, 3.11172591497380599197665768083, 3.52171657283372486042791577063, 4.70504402985700430590900760308, 5.90595360239356029232971639699, 6.88624385590397830675148105260, 7.57997030413241311653512235614, 8.091719234186432676401245269371, 9.225746435169066419696851592349, 10.07039877519672116518083358172
