| L(s) = 1 | + (0.955 + 0.294i)2-s + (−1.04 + 2.65i)3-s + (0.826 + 0.563i)4-s + (2.44 − 0.369i)5-s + (−1.77 + 2.22i)6-s + (0.623 + 0.781i)8-s + (−3.74 − 3.47i)9-s + (2.44 + 0.369i)10-s + (1.31 − 1.22i)11-s + (−2.35 + 1.60i)12-s + (1.16 + 5.12i)13-s + (−1.56 + 6.87i)15-s + (0.365 + 0.930i)16-s + (0.466 + 6.21i)17-s + (−2.55 − 4.42i)18-s + (0.588 − 1.01i)19-s + ⋯ |
| L(s) = 1 | + (0.675 + 0.208i)2-s + (−0.600 + 1.53i)3-s + (0.413 + 0.281i)4-s + (1.09 − 0.165i)5-s + (−0.724 + 0.908i)6-s + (0.220 + 0.276i)8-s + (−1.24 − 1.15i)9-s + (0.774 + 0.116i)10-s + (0.396 − 0.368i)11-s + (−0.679 + 0.463i)12-s + (0.324 + 1.42i)13-s + (−0.405 + 1.77i)15-s + (0.0913 + 0.232i)16-s + (0.113 + 1.50i)17-s + (−0.602 − 1.04i)18-s + (0.135 − 0.233i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.486 - 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10738 + 1.88409i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10738 + 1.88409i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.955 - 0.294i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (1.04 - 2.65i)T + (-2.19 - 2.04i)T^{2} \) |
| 5 | \( 1 + (-2.44 + 0.369i)T + (4.77 - 1.47i)T^{2} \) |
| 11 | \( 1 + (-1.31 + 1.22i)T + (0.822 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-1.16 - 5.12i)T + (-11.7 + 5.64i)T^{2} \) |
| 17 | \( 1 + (-0.466 - 6.21i)T + (-16.8 + 2.53i)T^{2} \) |
| 19 | \( 1 + (-0.588 + 1.01i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.260 + 3.47i)T + (-22.7 - 3.42i)T^{2} \) |
| 29 | \( 1 + (-0.395 - 0.190i)T + (18.0 + 22.6i)T^{2} \) |
| 31 | \( 1 + (5.26 + 9.11i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.553 - 0.377i)T + (13.5 - 34.4i)T^{2} \) |
| 41 | \( 1 + (3.47 + 4.35i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-0.256 + 0.322i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (5.88 + 1.81i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (-10.3 - 7.04i)T + (19.3 + 49.3i)T^{2} \) |
| 59 | \( 1 + (-5.81 - 0.876i)T + (56.3 + 17.3i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 2.37i)T + (22.2 - 56.7i)T^{2} \) |
| 67 | \( 1 + (0.229 + 0.396i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-9.78 + 4.71i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-10.0 + 3.10i)T + (60.3 - 41.1i)T^{2} \) |
| 79 | \( 1 + (1.29 - 2.25i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.80 - 7.89i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (3.60 + 3.34i)T + (6.65 + 88.7i)T^{2} \) |
| 97 | \( 1 + 12.3T + 97T^{2} \) |
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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
−10.82364849177828216172362440433, −9.926630128409589618067740547560, −9.270298728035640818630885304650, −8.459239020366468484165747445465, −6.73645792826723710654818210019, −5.97787358620636213410900141825, −5.38995804415941530383317971686, −4.27348116978728763257088677028, −3.73136559319056191780756106545, −2.01162340505127920810251977455,
1.07863774316458377853110090285, 2.13781593529218998016441100264, 3.24630835042082171512253540703, 5.24817586370817764897864794400, 5.57300959562904631996129753560, 6.63910814323813838005582869009, 7.15635122263011431410077890430, 8.185055770434843322936096136428, 9.534676707578646397376926741486, 10.35924416594392871438135053451
