| L(s) = 1 | + (0.142 − 0.989i)2-s + (0.415 + 0.909i)3-s + (−0.959 − 0.281i)4-s + (2.13 + 2.46i)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 + (2.74 − 1.76i)10-s + (−0.0420 − 0.292i)11-s + (−0.142 − 0.989i)12-s + (0.263 − 0.169i)13-s + (0.654 − 0.755i)14-s + (−1.35 + 2.97i)15-s + (0.841 + 0.540i)16-s + (4.27 − 1.25i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (0.239 + 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.956 + 1.10i)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 + (0.869 − 0.558i)10-s + (−0.0126 − 0.0881i)11-s + (−0.0410 − 0.285i)12-s + (0.0731 − 0.0469i)13-s + (0.175 − 0.201i)14-s + (−0.350 + 0.767i)15-s + (0.210 + 0.135i)16-s + (1.03 − 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.814 - 0.580i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96799 + 0.629237i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96799 + 0.629237i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-3.18 - 3.58i)T \) |
| good | 5 | \( 1 + (-2.13 - 2.46i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.0420 + 0.292i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.263 + 0.169i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.27 + 1.25i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (2.53 + 0.742i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (0.693 - 0.203i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.23 - 4.88i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.203 + 0.235i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-0.917 - 1.05i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.80 - 10.5i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + 12.8T + 47T^{2} \) |
| 53 | \( 1 + (0.340 + 0.218i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.85 + 3.11i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.78 + 6.09i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (0.222 - 1.54i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.425 + 2.96i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (-3.08 - 0.905i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-11.8 + 7.64i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (7.35 - 8.48i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (4.58 + 10.0i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (-3.71 - 4.29i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.977486708168013313850491915912, −9.670005646242268530322520822680, −8.691558019926482125234554494449, −7.67416656579267392661285851307, −6.56521963015554171628640741742, −5.63443410845049505703931583023, −4.81399507781401042851704015968, −3.44755547976921493255038958475, −2.80337045938586731407769001460, −1.66452418770840369803381204201,
0.988506356624495687702235330173, 2.18510156547673476071358362283, 3.81449509728666775404459126116, 4.93586877521430452968500406640, 5.64263784435257799741605711111, 6.45287385037481776913577132839, 7.46521904309479222316971593649, 8.292257307035526943077781242127, 8.903289164814827661231771320571, 9.683745472491731409274989040948
