| L(s) = 1 | + (−0.951 + 0.309i)2-s + (1.39 + 1.02i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−1.64 − 0.544i)6-s + (0.359 + 3.42i)7-s + (−0.587 + 0.809i)8-s + (0.896 + 2.86i)9-s + (0.669 − 0.743i)10-s + (−4.28 − 1.90i)11-s + (1.73 + 0.00929i)12-s + (0.215 + 1.01i)13-s + (−1.40 − 3.14i)14-s + (−1.72 − 0.190i)15-s + (0.309 − 0.951i)16-s + (−4.82 + 2.14i)17-s + ⋯ |
| L(s) = 1 | + (−0.672 + 0.218i)2-s + (0.805 + 0.592i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.671 − 0.222i)6-s + (0.136 + 1.29i)7-s + (−0.207 + 0.286i)8-s + (0.298 + 0.954i)9-s + (0.211 − 0.235i)10-s + (−1.29 − 0.574i)11-s + (0.499 + 0.00268i)12-s + (0.0596 + 0.280i)13-s + (−0.374 − 0.840i)14-s + (−0.444 − 0.0491i)15-s + (0.0772 − 0.237i)16-s + (−1.16 + 0.520i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 - 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0572193 + 0.854173i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0572193 + 0.854173i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-1.39 - 1.02i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (5.07 + 2.28i)T \) |
| good | 7 | \( 1 + (-0.359 - 3.42i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (4.28 + 1.90i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.215 - 1.01i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (4.82 - 2.14i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-1.67 - 0.356i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.800 - 0.581i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (2.44 + 7.52i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (2.37 + 1.36i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.93 - 5.34i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (-0.0691 + 0.325i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (-5.39 - 1.75i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (1.22 - 11.6i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (2.64 - 2.38i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 1.89iT - 61T^{2} \) |
| 67 | \( 1 + (-1.43 - 2.48i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (10.5 + 1.10i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (-1.87 + 4.20i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.71 - 10.5i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-4.44 + 4.93i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-7.36 + 5.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (-4.67 + 3.39i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.43688484676149926567743695408, −9.269387172362716906899050877661, −8.943977271601194400057156700653, −8.018082618216618273995222676745, −7.55752315175759425051184879727, −6.13207909708197995857734752252, −5.34387090600837712933504603787, −4.17115881527964229321266075920, −2.83889331589553022170391512739, −2.17049438479699618818032551941,
0.42585562659848665467737347677, 1.82521034863152491357981864295, 3.00452370220352190336498789970, 4.01620515596731044336230203484, 5.15936179209111077825413866296, 6.87174774708262252025842065680, 7.28791695767645309382014770607, 7.923122936614414259223042817324, 8.785848807115859315543575849515, 9.538010642689356490284864705064
