| L(s) = 1 | + (0.309 − 0.951i)2-s + (−0.669 + 0.743i)3-s + (−0.809 − 0.587i)4-s + (0.5 − 0.866i)5-s + (0.499 + 0.866i)6-s + (−0.426 + 4.05i)7-s + (−0.809 + 0.587i)8-s + (−0.104 − 0.994i)9-s + (−0.669 − 0.743i)10-s + (0.249 − 0.110i)11-s + (0.978 − 0.207i)12-s + (0.982 + 0.208i)13-s + (3.72 + 1.65i)14-s + (0.309 + 0.951i)15-s + (0.309 + 0.951i)16-s + (−2.45 − 1.09i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.386 + 0.429i)3-s + (−0.404 − 0.293i)4-s + (0.223 − 0.387i)5-s + (0.204 + 0.353i)6-s + (−0.161 + 1.53i)7-s + (−0.286 + 0.207i)8-s + (−0.0348 − 0.331i)9-s + (−0.211 − 0.235i)10-s + (0.0750 − 0.0334i)11-s + (0.282 − 0.0600i)12-s + (0.272 + 0.0579i)13-s + (0.996 + 0.443i)14-s + (0.0797 + 0.245i)15-s + (0.0772 + 0.237i)16-s + (−0.595 − 0.265i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.163 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.484873 + 0.571835i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.484873 + 0.571835i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.309 + 0.951i)T \) |
| 3 | \( 1 + (0.669 - 0.743i)T \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 + (3.82 - 4.04i)T \) |
| good | 7 | \( 1 + (0.426 - 4.05i)T + (-6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (-0.249 + 0.110i)T + (7.36 - 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.982 - 0.208i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (2.45 + 1.09i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (8.10 - 1.72i)T + (17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (4.05 - 2.94i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.32 + 4.08i)T + (-23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (-5.91 - 10.2i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.53 - 2.81i)T + (-4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (7.92 - 1.68i)T + (39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-1.12 - 3.46i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-0.479 - 4.56i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (2.45 - 2.72i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 6.54T + 61T^{2} \) |
| 67 | \( 1 + (-2.37 + 4.11i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.0675 + 0.642i)T + (-69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (1.41 - 0.627i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (1.28 + 0.570i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.748 - 0.830i)T + (-8.67 + 82.5i)T^{2} \) |
| 89 | \( 1 + (9.49 + 6.89i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (4.06 + 2.95i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.26352865516582878447375177624, −9.561404232220761418870964100069, −8.797656846053774121841752973344, −8.212020803760795922480259459192, −6.39399016002088171016274290569, −5.92724482209597053524890453976, −4.93874944166279768000620694885, −4.11551610884985184552006497844, −2.81052337102465999086069237412, −1.78847628432910814908302033269,
0.32822743391855199226590032750, 2.13584748875322415130942623119, 3.82250476029837494159083606927, 4.41266677932526412581196654613, 5.73154964614258892998931855513, 6.62175311446088390339135845826, 6.99338835590836998962518289762, 7.949783604576070587622275168925, 8.786509087931737356479981255163, 9.975792793627565241826048447851
