| L(s) = 1 | + (0.142 − 0.989i)2-s + (−0.415 − 0.909i)3-s + (−0.959 − 0.281i)4-s + (1.52 + 1.76i)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 + (1.96 − 1.26i)10-s + (−0.567 − 3.94i)11-s + (0.142 + 0.989i)12-s + (−1.78 + 1.15i)13-s + (0.654 − 0.755i)14-s + (0.968 − 2.11i)15-s + (0.841 + 0.540i)16-s + (0.646 − 0.189i)17-s + ⋯ |
| L(s) = 1 | + (0.100 − 0.699i)2-s + (−0.239 − 0.525i)3-s + (−0.479 − 0.140i)4-s + (0.682 + 0.787i)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 + (0.619 − 0.398i)10-s + (−0.171 − 1.18i)11-s + (0.0410 + 0.285i)12-s + (−0.496 + 0.318i)13-s + (0.175 − 0.201i)14-s + (0.249 − 0.547i)15-s + (0.210 + 0.135i)16-s + (0.156 − 0.0460i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.237 + 0.971i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33562 - 1.04830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33562 - 1.04830i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.51 + 1.62i)T \) |
| good | 5 | \( 1 + (-1.52 - 1.76i)T + (-0.711 + 4.94i)T^{2} \) |
| 11 | \( 1 + (0.567 + 3.94i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (1.78 - 1.15i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (-0.646 + 0.189i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-4.89 - 1.43i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.73 + 0.801i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (-3.12 + 6.85i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.44 + 7.44i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (-5.38 - 6.21i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 - 3.15i)T + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 1.49T + 47T^{2} \) |
| 53 | \( 1 + (5.43 + 3.49i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (-5.24 + 3.37i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (4.87 - 10.6i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (1.24 - 8.68i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (-2.21 + 15.3i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (11.3 + 3.32i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (11.3 - 7.26i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-4.03 + 4.65i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (-1.30 - 2.86i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (10.6 + 12.2i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−9.941916873532203789945576390487, −9.225164750426444075539107141124, −8.169382381203012564623692485868, −7.34071958181399490706317591747, −6.18031739059303511698273873950, −5.67759165614068065638716970702, −4.50951880880880840672694430789, −3.04829338469613648534278645332, −2.41885620946764776437689977974, −0.979266680832526305646413396410,
1.23266760449362162641587439862, 2.94380238064759529813010315365, 4.44991098999309191496603970099, 5.01776577466211121759562882184, 5.61608632205782695244816105577, 6.85984368120944991503936893508, 7.56869571315503038295171095801, 8.596736798411460359446998029340, 9.459294542136633164737107085886, 9.875937018285667043167354193218
