| L(s) = 1 | + (−2.06 + 0.217i)2-s + (−0.138 − 0.125i)3-s + (2.26 − 0.480i)4-s + (−1.04 − 2.34i)5-s + (0.313 + 0.228i)6-s + (2.62 − 0.288i)7-s + (−0.613 + 0.199i)8-s + (−0.309 − 2.94i)9-s + (2.66 + 4.61i)10-s + (0.754 − 3.22i)11-s + (−0.373 − 0.215i)12-s + (−0.672 + 0.488i)13-s + (−5.36 + 1.16i)14-s + (−0.148 + 0.455i)15-s + (−2.99 + 1.33i)16-s + (−0.606 + 5.77i)17-s + ⋯ |
| L(s) = 1 | + (−1.46 + 0.153i)2-s + (−0.0801 − 0.0721i)3-s + (1.13 − 0.240i)4-s + (−0.466 − 1.04i)5-s + (0.128 + 0.0930i)6-s + (0.994 − 0.109i)7-s + (−0.217 + 0.0705i)8-s + (−0.103 − 0.982i)9-s + (0.842 + 1.45i)10-s + (0.227 − 0.973i)11-s + (−0.107 − 0.0623i)12-s + (−0.186 + 0.135i)13-s + (−1.43 + 0.311i)14-s + (−0.0382 + 0.117i)15-s + (−0.749 + 0.333i)16-s + (−0.147 + 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.527 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.405642 - 0.225644i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.405642 - 0.225644i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.62 + 0.288i)T \) |
| 11 | \( 1 + (-0.754 + 3.22i)T \) |
| good | 2 | \( 1 + (2.06 - 0.217i)T + (1.95 - 0.415i)T^{2} \) |
| 3 | \( 1 + (0.138 + 0.125i)T + (0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (1.04 + 2.34i)T + (-3.34 + 3.71i)T^{2} \) |
| 13 | \( 1 + (0.672 - 0.488i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.606 - 5.77i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (0.316 + 0.0672i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-0.269 + 0.466i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.640 + 0.208i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.94 + 4.37i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-5.39 - 5.99i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-3.11 - 9.57i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.21iT - 43T^{2} \) |
| 47 | \( 1 + (2.31 - 10.8i)T + (-42.9 - 19.1i)T^{2} \) |
| 53 | \( 1 + (-2.09 - 0.932i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (2.64 + 12.4i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (6.78 - 3.01i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-2.07 - 3.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-10.1 - 7.35i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (12.0 - 2.56i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (-6.28 + 0.660i)T + (77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-2.52 - 1.83i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-5.37 - 3.10i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.779 - 1.07i)T + (-29.9 + 92.2i)T^{2} \) |
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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
−14.60926375075632187750762099941, −13.05708874561615585211719172249, −11.79222642948854527472815090040, −10.92985498647616154105387539393, −9.486575073577234328859662510296, −8.503454567645344765811094304422, −7.967581087444428446205325262744, −6.26278204138012873726186546407, −4.33131451322708136340620317105, −1.10336413588093118949557414938,
2.29153844689280663270348084750, 4.83669420232159345853382566495, 7.19208505771094422510967497294, 7.70103476983337795671206742875, 9.048240366588116472459243968375, 10.32127277337528961807428435760, 11.03528160095999354509178956070, 11.87490086968280678654146200937, 13.82787241474278581045028128298, 14.82263607574101460397436311611
