| L(s) = 1 | + (−0.669 + 0.743i)2-s + (−0.913 + 0.406i)3-s + (−0.104 − 0.994i)4-s + (2.51 − 0.534i)5-s + (0.309 − 0.951i)6-s + (1.73 + 1.99i)7-s + (0.809 + 0.587i)8-s + (0.669 − 0.743i)9-s + (−1.28 + 2.22i)10-s + (0.623 − 3.25i)11-s + (0.5 + 0.866i)12-s + (−1.70 − 5.23i)13-s + (−2.64 − 0.0451i)14-s + (−2.07 + 1.51i)15-s + (−0.978 + 0.207i)16-s + (1.46 + 1.62i)17-s + ⋯ |
| L(s) = 1 | + (−0.473 + 0.525i)2-s + (−0.527 + 0.234i)3-s + (−0.0522 − 0.497i)4-s + (1.12 − 0.238i)5-s + (0.126 − 0.388i)6-s + (0.656 + 0.754i)7-s + (0.286 + 0.207i)8-s + (0.223 − 0.247i)9-s + (−0.406 + 0.703i)10-s + (0.187 − 0.982i)11-s + (0.144 + 0.249i)12-s + (−0.471 − 1.45i)13-s + (−0.707 − 0.0120i)14-s + (−0.536 + 0.390i)15-s + (−0.244 + 0.0519i)16-s + (0.355 + 0.394i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 - 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.19132 + 0.252572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.19132 + 0.252572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.669 - 0.743i)T \) |
| 3 | \( 1 + (0.913 - 0.406i)T \) |
| 7 | \( 1 + (-1.73 - 1.99i)T \) |
| 11 | \( 1 + (-0.623 + 3.25i)T \) |
| good | 5 | \( 1 + (-2.51 + 0.534i)T + (4.56 - 2.03i)T^{2} \) |
| 13 | \( 1 + (1.70 + 5.23i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-1.46 - 1.62i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-0.269 + 2.56i)T + (-18.5 - 3.95i)T^{2} \) |
| 23 | \( 1 + (-3.88 - 6.72i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-6.36 + 4.62i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-4.90 - 1.04i)T + (28.3 + 12.6i)T^{2} \) |
| 37 | \( 1 + (8.81 + 3.92i)T + (24.7 + 27.4i)T^{2} \) |
| 41 | \( 1 + (-5.19 - 3.77i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 - 2.45T + 43T^{2} \) |
| 47 | \( 1 + (1.27 - 12.0i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-5.17 - 1.09i)T + (48.4 + 21.5i)T^{2} \) |
| 59 | \( 1 + (0.721 + 6.86i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + (0.523 - 0.111i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (6.99 - 12.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.58 - 4.87i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (1.21 + 11.5i)T + (-71.4 + 15.1i)T^{2} \) |
| 79 | \( 1 + (-1.38 + 1.53i)T + (-8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-2.41 + 7.43i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (8.51 + 14.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.41 + 7.43i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−10.92013866395860225813416343376, −10.11910415319732956577113868760, −9.301660707420510919913552960499, −8.493279515400003589635066144258, −7.54405446397210688702796895661, −6.08698237533878709558835854841, −5.65958229836976066449200185504, −4.87348886371213534682150572647, −2.86000939171952351485724008341, −1.18087178676279938492833048172,
1.38743460516600492406858588359, 2.37907603369048642758543375404, 4.27111988308292838057035629197, 5.12861468702078362761676752851, 6.70540186831429986850678215935, 7.05397304804918946422541696099, 8.410811698486421310161111317997, 9.474671318090839139125718112960, 10.22965912428201151831962010931, 10.74286857611029529228374212209
