| L(s) = 1 | + (−1.21 − 0.722i)2-s + (0.509 + 1.11i)3-s + (0.956 + 1.75i)4-s + (−0.654 − 0.755i)5-s + (0.186 − 1.72i)6-s + (−4.15 − 2.67i)7-s + (0.105 − 2.82i)8-s + (0.980 − 1.13i)9-s + (0.250 + 1.39i)10-s + (−2.60 + 0.374i)11-s + (−1.47 + 1.96i)12-s + (2.34 + 3.65i)13-s + (3.12 + 6.24i)14-s + (0.509 − 1.11i)15-s + (−2.16 + 3.36i)16-s + (−0.801 − 2.72i)17-s + ⋯ |
| L(s) = 1 | + (−0.859 − 0.510i)2-s + (0.294 + 0.643i)3-s + (0.478 + 0.878i)4-s + (−0.292 − 0.337i)5-s + (0.0760 − 0.703i)6-s + (−1.57 − 1.00i)7-s + (0.0372 − 0.999i)8-s + (0.326 − 0.377i)9-s + (0.0791 + 0.440i)10-s + (−0.784 + 0.112i)11-s + (−0.424 + 0.566i)12-s + (0.651 + 1.01i)13-s + (0.834 + 1.66i)14-s + (0.131 − 0.287i)15-s + (−0.542 + 0.840i)16-s + (−0.194 − 0.662i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.563637 + 0.349868i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.563637 + 0.349868i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 + 0.722i)T \) |
| 5 | \( 1 + (0.654 + 0.755i)T \) |
| 23 | \( 1 + (-4.08 - 2.51i)T \) |
| good | 3 | \( 1 + (-0.509 - 1.11i)T + (-1.96 + 2.26i)T^{2} \) |
| 7 | \( 1 + (4.15 + 2.67i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (2.60 - 0.374i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.34 - 3.65i)T + (-5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (0.801 + 2.72i)T + (-14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (2.20 - 7.49i)T + (-15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-1.81 - 6.17i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-2.98 - 1.36i)T + (20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-5.70 + 6.57i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-2.73 - 3.15i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (5.19 - 2.37i)T + (28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 2.74iT - 47T^{2} \) |
| 53 | \( 1 + (1.12 + 0.721i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-1.31 + 0.847i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (5.57 - 12.2i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (4.54 + 0.653i)T + (64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (8.12 + 1.16i)T + (68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-0.535 - 0.157i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-1.15 + 0.740i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-5.77 - 5.00i)T + (11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-9.30 + 4.24i)T + (58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (12.7 - 11.0i)T + (13.8 - 96.0i)T^{2} \) |
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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
−10.23233418413560845388451576318, −9.371922010577123534867693928784, −8.983488287575555373589746005660, −7.80331067312724645322153839737, −7.00342330245748920243625846216, −6.24998561983328416733283918496, −4.43438934054760391161722059894, −3.71658500338093710100597656204, −2.99447606309669871505069549208, −1.16636882752809209528646186989,
0.45355834833231618797471565415, 2.41070020538771709842852170008, 2.96981147250114386511404347474, 4.88845168153060779040838953183, 6.11991046137808676815734463872, 6.52204033153291304071898553704, 7.45479472987254517853382969459, 8.324307068706337266037589032939, 8.839585966906168171490282931956, 9.900532223388321460878557260059
