| L(s) = 1 | + (−1.73 − 0.368i)2-s + (0.324 − 3.08i)3-s + (1.04 + 0.467i)4-s + (−0.669 − 0.743i)5-s + (−1.70 + 5.23i)6-s + (0.427 + 2.61i)7-s + (1.22 + 0.887i)8-s + (−6.49 − 1.37i)9-s + (0.887 + 1.53i)10-s + (−2.85 + 1.69i)11-s + (1.78 − 3.08i)12-s + (1.56 + 4.81i)13-s + (0.221 − 4.69i)14-s + (−2.51 + 1.82i)15-s + (−3.33 − 3.69i)16-s + (−6.63 + 1.41i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.260i)2-s + (0.187 − 1.78i)3-s + (0.524 + 0.233i)4-s + (−0.299 − 0.332i)5-s + (−0.695 + 2.13i)6-s + (0.161 + 0.986i)7-s + (0.431 + 0.313i)8-s + (−2.16 − 0.459i)9-s + (0.280 + 0.486i)10-s + (−0.859 + 0.510i)11-s + (0.514 − 0.891i)12-s + (0.434 + 1.33i)13-s + (0.0592 − 1.25i)14-s + (−0.648 + 0.471i)15-s + (−0.832 − 0.924i)16-s + (−1.61 + 0.342i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.325 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0898188 + 0.0640995i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0898188 + 0.0640995i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (0.669 + 0.743i)T \) |
| 7 | \( 1 + (-0.427 - 2.61i)T \) |
| 11 | \( 1 + (2.85 - 1.69i)T \) |
| good | 2 | \( 1 + (1.73 + 0.368i)T + (1.82 + 0.813i)T^{2} \) |
| 3 | \( 1 + (-0.324 + 3.08i)T + (-2.93 - 0.623i)T^{2} \) |
| 13 | \( 1 + (-1.56 - 4.81i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (6.63 - 1.41i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (4.11 - 1.83i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.37 + 2.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.53 + 1.11i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-1.69 + 1.88i)T + (-3.24 - 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.227 - 2.16i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (7.04 + 5.11i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.20T + 43T^{2} \) |
| 47 | \( 1 + (-2.48 + 1.10i)T + (31.4 - 34.9i)T^{2} \) |
| 53 | \( 1 + (0.972 - 1.07i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (8.37 + 3.72i)T + (39.4 + 43.8i)T^{2} \) |
| 61 | \( 1 + (4.49 + 4.99i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-4.25 - 7.37i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.762 - 2.34i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.291 - 0.129i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (-4.77 - 1.01i)T + (72.1 + 32.1i)T^{2} \) |
| 83 | \( 1 + (-5.48 + 16.8i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (0.447 - 0.775i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.39 + 4.28i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.56959116021656918871797491364, −10.71895949110513228941341932305, −9.246464730887871483596663810001, −8.534115018500658742265108900754, −8.175038355049660087013594266377, −7.03411266359841497978299882722, −6.26840295766669829628473764044, −4.72404612798597850964653860831, −2.32420733713561503082710886442, −1.77475117643829163360657939505,
0.10236038680233372997130179584, 3.03081972507102954418803258200, 4.15136052062025672088436454976, 5.03709637367144483760151985269, 6.61688777643655958769021390578, 7.910471146689938351127464382461, 8.490270948317607393333388278946, 9.345706349600611937449468352118, 10.36558403850733690826373751952, 10.69977713833649088105839889117
