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
−9.875937018285667043167354193218, −9.459294542136633164737107085886, −8.596736798411460359446998029340, −7.56869571315503038295171095801, −6.85984368120944991503936893508, −5.61608632205782695244816105577, −5.01776577466211121759562882184, −4.44991098999309191496603970099, −2.94380238064759529813010315365, −1.23266760449362162641587439862,
0.979266680832526305646413396410, 2.41885620946764776437689977974, 3.04829338469613648534278645332, 4.50951880880880840672694430789, 5.67759165614068065638716970702, 6.18031739059303511698273873950, 7.34071958181399490706317591747, 8.169382381203012564623692485868, 9.225164750426444075539107141124, 9.941916873532203789945576390487