| L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.567 + 1.63i)3-s + (0.809 − 0.587i)4-s + (−0.866 + 0.5i)5-s + (−0.0340 + 1.73i)6-s + (0.503 + 4.79i)7-s + (0.587 − 0.809i)8-s + (−2.35 − 1.85i)9-s + (−0.669 + 0.743i)10-s + (−3.34 − 1.48i)11-s + (0.502 + 1.65i)12-s + (0.191 + 0.900i)13-s + (1.96 + 4.40i)14-s + (−0.326 − 1.70i)15-s + (0.309 − 0.951i)16-s + (−3.22 + 1.43i)17-s + ⋯ |
| L(s) = 1 | + (0.672 − 0.218i)2-s + (−0.327 + 0.944i)3-s + (0.404 − 0.293i)4-s + (−0.387 + 0.223i)5-s + (−0.0138 + 0.706i)6-s + (0.190 + 1.81i)7-s + (0.207 − 0.286i)8-s + (−0.785 − 0.619i)9-s + (−0.211 + 0.235i)10-s + (−1.00 − 0.449i)11-s + (0.145 + 0.478i)12-s + (0.0530 + 0.249i)13-s + (0.523 + 1.17i)14-s + (−0.0843 − 0.439i)15-s + (0.0772 − 0.237i)16-s + (−0.782 + 0.348i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.235224 + 1.12520i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.235224 + 1.12520i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.567 - 1.63i)T \) |
| 5 | \( 1 + (0.866 - 0.5i)T \) |
| 31 | \( 1 + (-4.68 + 3.00i)T \) |
| good | 7 | \( 1 + (-0.503 - 4.79i)T + (-6.84 + 1.45i)T^{2} \) |
| 11 | \( 1 + (3.34 + 1.48i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (-0.191 - 0.900i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (3.22 - 1.43i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (2.91 + 0.618i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.0893 - 0.0649i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.02 - 3.14i)T + (-23.4 + 17.0i)T^{2} \) |
| 37 | \( 1 + (0.546 + 0.315i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.90 + 7.12i)T + (4.28 + 40.7i)T^{2} \) |
| 43 | \( 1 + (2.30 - 10.8i)T + (-39.2 - 17.4i)T^{2} \) |
| 47 | \( 1 + (1.38 + 0.451i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (0.987 - 9.39i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (0.439 - 0.395i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 - 6.64iT - 61T^{2} \) |
| 67 | \( 1 + (-6.53 - 11.3i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-11.4 - 1.19i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (0.190 - 0.426i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-1.63 - 3.66i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.02 + 6.69i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (8.73 - 6.34i)T + (27.5 - 84.6i)T^{2} \) |
| 97 | \( 1 + (9.20 - 6.68i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.68794236099072919207531831505, −9.709202699747808253975871925496, −8.734042476938716812310920102368, −8.233359713240115882071344544607, −6.62823097336267870936780579092, −5.84719466893956655057830051745, −5.14786826683467807784252619572, −4.32893392954284477092112886285, −3.09952821955749675981955060146, −2.34589020977447949967702135200,
0.42036559327047420960965316258, 1.97662105877848830208386103602, 3.37618274777825013615086915336, 4.54817203194411704431273696778, 5.13634385535994993261050318382, 6.57647485244788377291832970859, 6.96075512305889848132321242778, 7.915251736423673754276997345078, 8.282627497568380184337397166753, 10.01957916748455962775575748681
