| L(s) = 1 | + (0.329 + 0.380i)3-s + (−0.142 − 0.989i)5-s + (−0.847 + 1.85i)7-s + (0.390 − 2.71i)9-s + (4.95 + 1.45i)11-s + (0.773 + 1.69i)13-s + (0.329 − 0.380i)15-s + (0.687 + 0.441i)17-s + (5.37 − 3.45i)19-s + (−0.983 + 0.288i)21-s + (4.01 + 2.62i)23-s + (−0.959 + 0.281i)25-s + (2.43 − 1.56i)27-s + (−1.82 − 1.17i)29-s + (−2.33 + 2.69i)31-s + ⋯ |
| L(s) = 1 | + (0.190 + 0.219i)3-s + (−0.0636 − 0.442i)5-s + (−0.320 + 0.701i)7-s + (0.130 − 0.906i)9-s + (1.49 + 0.438i)11-s + (0.214 + 0.469i)13-s + (0.0850 − 0.0981i)15-s + (0.166 + 0.107i)17-s + (1.23 − 0.792i)19-s + (−0.214 + 0.0630i)21-s + (0.837 + 0.546i)23-s + (−0.191 + 0.0563i)25-s + (0.467 − 0.300i)27-s + (−0.339 − 0.218i)29-s + (−0.418 + 0.483i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.58518 + 0.0892890i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.58518 + 0.0892890i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 23 | \( 1 + (-4.01 - 2.62i)T \) |
| good | 3 | \( 1 + (-0.329 - 0.380i)T + (-0.426 + 2.96i)T^{2} \) |
| 7 | \( 1 + (0.847 - 1.85i)T + (-4.58 - 5.29i)T^{2} \) |
| 11 | \( 1 + (-4.95 - 1.45i)T + (9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-0.773 - 1.69i)T + (-8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-0.687 - 0.441i)T + (7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-5.37 + 3.45i)T + (7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (1.82 + 1.17i)T + (12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (2.33 - 2.69i)T + (-4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (0.0791 - 0.550i)T + (-35.5 - 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.207 - 1.44i)T + (-39.3 + 11.5i)T^{2} \) |
| 43 | \( 1 + (6.92 + 7.99i)T + (-6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 1.60T + 47T^{2} \) |
| 53 | \( 1 + (2.53 - 5.55i)T + (-34.7 - 40.0i)T^{2} \) |
| 59 | \( 1 + (2.28 + 5.00i)T + (-38.6 + 44.5i)T^{2} \) |
| 61 | \( 1 + (0.676 - 0.780i)T + (-8.68 - 60.3i)T^{2} \) |
| 67 | \( 1 + (-4.15 + 1.21i)T + (56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (0.396 - 0.116i)T + (59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (5.14 - 3.30i)T + (30.3 - 66.4i)T^{2} \) |
| 79 | \( 1 + (4.50 + 9.85i)T + (-51.7 + 59.7i)T^{2} \) |
| 83 | \( 1 + (2.02 - 14.0i)T + (-79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (8.47 + 9.78i)T + (-12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (2.60 + 18.0i)T + (-93.0 + 27.3i)T^{2} \) |
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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
−11.33348379505762691267532829246, −9.815011626333114357009286723007, −9.190700498207501199396558638299, −8.803662831197264527159382493046, −7.25556433320997246633649072429, −6.47956839593417822708784106825, −5.35712101125630921913746778720, −4.14435028875464908332149718966, −3.17458932373806496592572073821, −1.36591035411896526019243091749,
1.33152744911835822783556703826, 3.07194414602431296857114043491, 4.00765458242682506291246334354, 5.39881965935053457804393843865, 6.56283819843668554145461055640, 7.33959038472900328620342759883, 8.199213565952537524389713441462, 9.313363779174760592485412696267, 10.19410053476430970026395935693, 11.03225337464687917586205305615
