| L(s) = 1 | + (1.40 + 0.128i)2-s + (0.819 − 0.573i)3-s + (1.96 + 0.360i)4-s + (1.82 − 0.160i)5-s + (1.22 − 0.702i)6-s + (1.50 + 2.61i)7-s + (2.72 + 0.760i)8-s + (0.342 − 0.939i)9-s + (2.59 + 0.00898i)10-s + (−1.39 + 0.375i)11-s + (1.81 − 0.832i)12-s + (−0.659 − 0.462i)13-s + (1.78 + 3.87i)14-s + (1.40 − 1.18i)15-s + (3.73 + 1.41i)16-s + (−4.63 + 1.68i)17-s + ⋯ |
| L(s) = 1 | + (0.995 + 0.0906i)2-s + (0.472 − 0.331i)3-s + (0.983 + 0.180i)4-s + (0.818 − 0.0715i)5-s + (0.500 − 0.286i)6-s + (0.569 + 0.986i)7-s + (0.963 + 0.268i)8-s + (0.114 − 0.313i)9-s + (0.821 + 0.00284i)10-s + (−0.422 + 0.113i)11-s + (0.524 − 0.240i)12-s + (−0.183 − 0.128i)13-s + (0.477 + 1.03i)14-s + (0.363 − 0.304i)15-s + (0.934 + 0.354i)16-s + (−1.12 + 0.409i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 - 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.95928 + 0.297996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.95928 + 0.297996i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.40 - 0.128i)T \) |
| 3 | \( 1 + (-0.819 + 0.573i)T \) |
| 19 | \( 1 + (1.84 + 3.95i)T \) |
| good | 5 | \( 1 + (-1.82 + 0.160i)T + (4.92 - 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.50 - 2.61i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.39 - 0.375i)T + (9.52 - 5.5i)T^{2} \) |
| 13 | \( 1 + (0.659 + 0.462i)T + (4.44 + 12.2i)T^{2} \) |
| 17 | \( 1 + (4.63 - 1.68i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (-0.682 + 0.572i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (5.81 - 2.71i)T + (18.6 - 22.2i)T^{2} \) |
| 31 | \( 1 + (-1.18 - 2.04i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.83 + 3.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (0.0654 - 0.370i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (1.03 + 11.8i)T + (-42.3 + 7.46i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 5.97i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (-4.72 - 0.413i)T + (52.1 + 9.20i)T^{2} \) |
| 59 | \( 1 + (-0.550 + 1.18i)T + (-37.9 - 45.1i)T^{2} \) |
| 61 | \( 1 + (-6.67 - 0.584i)T + (60.0 + 10.5i)T^{2} \) |
| 67 | \( 1 + (-0.0913 - 0.195i)T + (-43.0 + 51.3i)T^{2} \) |
| 71 | \( 1 + (-0.592 + 0.705i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-6.83 - 1.20i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (-1.25 + 7.11i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (5.21 + 1.39i)T + (71.8 + 41.5i)T^{2} \) |
| 89 | \( 1 + (-0.935 - 5.30i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (-5.66 - 15.5i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−10.27636115820370212601072045249, −9.037729567371206088523121479876, −8.505936833776181413516311425104, −7.35600117987812623797094653012, −6.59158352778383414390543997058, −5.57945341260633206688474492791, −5.03155911444948566488376166605, −3.77266689017179905510157557702, −2.34570560991575362077415289084, −2.03647041771255477570054671890,
1.67461915863097859960645619390, 2.63179635809806860687877254339, 3.91064449405692376528257183612, 4.59194522843977430989310894872, 5.57355411900419027083782293621, 6.50443182545616015111122749531, 7.45222620726216443984148384927, 8.203401679738770052037577500665, 9.521068910975470903893703563591, 10.17854328547754806812890933192
