| L(s) = 1 | + (−1.06 + 1.26i)2-s + (−0.124 − 0.708i)4-s + (0.908 + 2.04i)5-s + (−1.22 − 0.215i)7-s + (−1.82 − 1.05i)8-s + (−3.54 − 1.01i)10-s + (−3.30 + 1.20i)11-s + (0.380 + 0.452i)13-s + (1.56 − 1.31i)14-s + (4.62 − 1.68i)16-s + (−4.46 + 2.58i)17-s + (−1.41 + 2.45i)19-s + (1.33 − 0.899i)20-s + (1.98 − 5.44i)22-s + (6.59 − 1.16i)23-s + ⋯ |
| L(s) = 1 | + (−0.749 + 0.893i)2-s + (−0.0624 − 0.354i)4-s + (0.406 + 0.913i)5-s + (−0.461 − 0.0814i)7-s + (−0.646 − 0.373i)8-s + (−1.12 − 0.321i)10-s + (−0.996 + 0.362i)11-s + (0.105 + 0.125i)13-s + (0.418 − 0.351i)14-s + (1.15 − 0.420i)16-s + (−1.08 + 0.625i)17-s + (−0.325 + 0.563i)19-s + (0.298 − 0.201i)20-s + (0.422 − 1.16i)22-s + (1.37 − 0.242i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 + 0.467i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.118779 - 0.479007i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.118779 - 0.479007i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.908 - 2.04i)T \) |
| good | 2 | \( 1 + (1.06 - 1.26i)T + (-0.347 - 1.96i)T^{2} \) |
| 7 | \( 1 + (1.22 + 0.215i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (3.30 - 1.20i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.380 - 0.452i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (4.46 - 2.58i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.41 - 2.45i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.59 + 1.16i)T + (21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (7.21 + 6.05i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.196 - 1.11i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.06 - 2.34i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.43 - 3.71i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.739 + 2.03i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (-6.14 - 1.08i)T + (44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 8.21iT - 53T^{2} \) |
| 59 | \( 1 + (-12.6 - 4.62i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (1.38 - 7.85i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.82 - 3.37i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-3.43 - 5.95i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-1.23 - 0.715i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.970 + 0.814i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.982 + 1.17i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (6.52 - 11.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-4.74 - 13.0i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.57161054625787540009113368667, −10.54844040940978622146899583952, −9.894751302183347900856780392353, −8.935715633732151295524623884872, −8.009280295490853749611071687058, −7.04033046006180228348111424995, −6.49419244791338749268001917918, −5.44624366135431851633156616020, −3.69940632101950726393854471845, −2.41042848911350088661301421326,
0.38243046484647628005294603000, 1.99824267027024864501717077332, 3.16497256626543504934249734048, 4.92437758942475245144229845854, 5.74414724996774045225108760386, 7.09298083477582588142078121147, 8.504860816383659855715432971797, 9.040790302701093966096383546304, 9.740874200833889422359950912810, 10.80667586341117200150159255906
