| L(s) = 1 | + (−1.77 + 0.917i)2-s + (−4.35 + 3.57i)3-s + (2.31 − 3.26i)4-s + (−6.55 − 1.98i)5-s + (4.46 − 10.3i)6-s + (−0.342 − 1.72i)7-s + (−1.12 + 7.92i)8-s + (4.43 − 22.2i)9-s + (13.4 − 2.48i)10-s + (1.65 + 16.7i)11-s + (1.55 + 22.4i)12-s + (1.07 + 3.52i)13-s + (2.18 + 2.74i)14-s + (35.6 − 14.7i)15-s + (−5.25 − 15.1i)16-s + (−1.32 + 3.18i)17-s + ⋯ |
| L(s) = 1 | + (−0.888 + 0.458i)2-s + (−1.45 + 1.19i)3-s + (0.579 − 0.815i)4-s + (−1.31 − 0.397i)5-s + (0.743 − 1.72i)6-s + (−0.0488 − 0.245i)7-s + (−0.141 + 0.990i)8-s + (0.492 − 2.47i)9-s + (1.34 − 0.248i)10-s + (0.150 + 1.52i)11-s + (0.129 + 1.87i)12-s + (0.0823 + 0.271i)13-s + (0.156 + 0.195i)14-s + (2.37 − 0.984i)15-s + (−0.328 − 0.944i)16-s + (−0.0776 + 0.187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.279121 - 0.0348450i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.279121 - 0.0348450i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.77 - 0.917i)T \) |
| good | 3 | \( 1 + (4.35 - 3.57i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (6.55 + 1.98i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.342 + 1.72i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-1.65 - 16.7i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (-1.07 - 3.52i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (1.32 - 3.18i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (-10.9 + 20.4i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-18.6 + 12.4i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (-4.01 + 40.7i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (20.9 + 20.9i)T + 961iT^{2} \) |
| 37 | \( 1 + (12.4 + 23.2i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (17.4 - 11.6i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (-36.4 - 29.8i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (12.8 - 31.0i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (5.83 + 59.2i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (73.1 + 22.1i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-23.6 + 19.4i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (15.2 + 18.5i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-102. + 20.2i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (-32.6 - 6.49i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-12.5 - 30.2i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (112. + 59.9i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (-45.5 + 68.2i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (11.4 - 11.4i)T - 9.40e3iT^{2} \) |
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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
−12.52197686731025869970486519226, −11.58100298925838837541209842102, −11.03353682921131469382334013729, −9.900461230598178777609544377347, −9.135633290751069567532042849099, −7.54793206925138465267743204269, −6.55602551186897361592252583219, −5.02673335847960399264947410150, −4.22572309483122242747954779731, −0.39579059872999036164079080149,
1.05572195599635207741560872096, 3.35117901198494931582820986728, 5.61277385205538618791041487493, 6.88854533905910131473567972520, 7.63730070070273747237372569061, 8.629573431363721733309132714700, 10.66939454226675357535735876603, 11.12791689592390501061648033647, 11.96036554112527126699072711524, 12.49401634494397712786812119234
