| L(s) = 1 | + (0.142 + 0.989i)2-s + (−0.415 + 0.909i)3-s + (−0.959 + 0.281i)4-s + (1.64 − 1.89i)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 + (2.10 + 1.35i)10-s + (0.480 − 3.34i)11-s + (0.142 − 0.989i)12-s + (0.299 + 0.192i)13-s + (−0.654 − 0.755i)14-s + (1.04 + 2.28i)15-s + (0.841 − 0.540i)16-s + (−5.90 − 1.73i)17-s + ⋯ |
| L(s) = 1 | + (0.100 + 0.699i)2-s + (−0.239 + 0.525i)3-s + (−0.479 + 0.140i)4-s + (0.734 − 0.847i)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 + (0.667 + 0.428i)10-s + (0.144 − 1.00i)11-s + (0.0410 − 0.285i)12-s + (0.0831 + 0.0534i)13-s + (−0.175 − 0.201i)14-s + (0.268 + 0.588i)15-s + (0.210 − 0.135i)16-s + (−1.43 − 0.420i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 + 0.0929i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.995 + 0.0929i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44738 - 0.0673797i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44738 - 0.0673797i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.948 + 4.70i)T \) |
| good | 5 | \( 1 + (-1.64 + 1.89i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.480 + 3.34i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (-0.299 - 0.192i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (5.90 + 1.73i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (-7.17 + 2.10i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-4.93 - 1.44i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (2.68 + 5.87i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (2.94 + 3.39i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-3.88 + 4.48i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-0.138 + 0.303i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 - 4.57T + 47T^{2} \) |
| 53 | \( 1 + (0.868 - 0.558i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (-4.45 - 2.86i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-5.21 - 11.4i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (-0.581 - 4.04i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (1.72 + 11.9i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-2.76 + 0.810i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (0.698 + 0.449i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (5.84 + 6.74i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (2.01 - 4.40i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (8.92 - 10.3i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−9.723564342658761358842819500249, −8.922870581997444026689027729211, −8.781262717728624264185356906013, −7.36566785904116658194864750064, −6.39857178150419094997709480496, −5.61501332625764775415397838228, −4.99393358105720970417156941660, −3.98791205032935635718196491515, −2.67345092900133493416340074469, −0.72149002261913543797052605064,
1.44146104357437530249590778392, 2.44959350683813394426825717988, 3.46116569690257496490151869997, 4.73316066337614224590051144123, 5.75260615497032491760316088245, 6.69608492891220667681779271145, 7.26000958434201071126972271627, 8.458937527674446786567415949737, 9.608782428500734895915303671102, 9.990360039799040213914488256032
