| L(s) = 1 | + (−0.142 + 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (−2.74 − 3.16i)5-s + (0.959 − 0.281i)6-s + (−0.841 − 0.540i)7-s + (0.415 − 0.909i)8-s + (−0.654 + 0.755i)9-s + (3.52 − 2.26i)10-s + (−0.677 − 4.71i)11-s + (0.142 + 0.989i)12-s + (1.33 − 0.856i)13-s + (0.654 − 0.755i)14-s + (−1.74 + 3.81i)15-s + (0.841 + 0.540i)16-s + (−6.38 + 1.87i)17-s + ⋯ |
| L(s) = 1 | + (−0.100 + 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (−1.22 − 1.41i)5-s + (0.391 − 0.115i)6-s + (−0.317 − 0.204i)7-s + (0.146 − 0.321i)8-s + (−0.218 + 0.251i)9-s + (1.11 − 0.717i)10-s + (−0.204 − 1.42i)11-s + (0.0410 + 0.285i)12-s + (0.369 − 0.237i)13-s + (0.175 − 0.201i)14-s + (−0.449 + 0.985i)15-s + (0.210 + 0.135i)16-s + (−1.54 + 0.454i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.861 - 0.507i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0346802 + 0.127199i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0346802 + 0.127199i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.142 - 0.989i)T \) |
| 3 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 23 | \( 1 + (1.98 - 4.36i)T \) |
| good | 5 | \( 1 + (2.74 + 3.16i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.677 + 4.71i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 0.856i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (6.38 - 1.87i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-3.12 - 0.917i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-9.16 + 2.69i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.35 + 2.97i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (7.59 - 8.76i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (2.50 + 2.89i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (1.52 + 3.33i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 0.164T + 47T^{2} \) |
| 53 | \( 1 + (-1.25 - 0.807i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (10.2 - 6.61i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-3.74 + 8.20i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.24 - 8.68i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (0.336 - 2.33i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-6.42 - 1.88i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (5.33 - 3.43i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.66 + 4.23i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.68 + 10.2i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.55 + 2.95i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.090191258789315163167404061246, −8.368391177146867037748270034116, −8.124609887865137927004603385870, −7.04986203127782217594913483678, −6.10895798032682760442030099468, −5.25112501698359310519753852241, −4.32038916390894295462352509028, −3.36428622230780794112402872473, −1.15050150457646789201562447541, −0.07562455741484300577797867408,
2.36149159245211550868431914933, 3.19971002112751462226017304625, 4.20527527301134503923043273274, 4.84766239605514615559816232442, 6.57799458072493475673771146145, 6.98051745237113782784616254028, 8.074318689231473699074479279048, 9.015649103642855430864109890286, 9.944469610227775835892622676086, 10.67135640516304272219533277010
