| L(s) = 1 | + (−0.687 + 1.87i)2-s + (3.91 − 3.21i)3-s + (−3.05 − 2.58i)4-s + (−8.46 − 2.56i)5-s + (3.34 + 9.55i)6-s + (−0.593 − 2.98i)7-s + (6.95 − 3.96i)8-s + (3.24 − 16.3i)9-s + (10.6 − 14.1i)10-s + (0.723 + 7.34i)11-s + (−20.2 − 0.296i)12-s + (−5.58 − 18.4i)13-s + (6.00 + 0.936i)14-s + (−41.3 + 17.1i)15-s + (2.66 + 15.7i)16-s + (0.326 − 0.787i)17-s + ⋯ |
| L(s) = 1 | + (−0.343 + 0.939i)2-s + (1.30 − 1.07i)3-s + (−0.763 − 0.645i)4-s + (−1.69 − 0.513i)5-s + (0.556 + 1.59i)6-s + (−0.0847 − 0.426i)7-s + (0.868 − 0.495i)8-s + (0.360 − 1.81i)9-s + (1.06 − 1.41i)10-s + (0.0657 + 0.667i)11-s + (−1.68 − 0.0246i)12-s + (−0.429 − 1.41i)13-s + (0.429 + 0.0668i)14-s + (−2.75 + 1.14i)15-s + (0.166 + 0.986i)16-s + (0.0191 − 0.0463i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.240 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.869687 - 0.680181i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.869687 - 0.680181i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.687 - 1.87i)T \) |
| good | 3 | \( 1 + (-3.91 + 3.21i)T + (1.75 - 8.82i)T^{2} \) |
| 5 | \( 1 + (8.46 + 2.56i)T + (20.7 + 13.8i)T^{2} \) |
| 7 | \( 1 + (0.593 + 2.98i)T + (-45.2 + 18.7i)T^{2} \) |
| 11 | \( 1 + (-0.723 - 7.34i)T + (-118. + 23.6i)T^{2} \) |
| 13 | \( 1 + (5.58 + 18.4i)T + (-140. + 93.8i)T^{2} \) |
| 17 | \( 1 + (-0.326 + 0.787i)T + (-204. - 204. i)T^{2} \) |
| 19 | \( 1 + (-3.73 + 6.98i)T + (-200. - 300. i)T^{2} \) |
| 23 | \( 1 + (-28.7 + 19.1i)T + (202. - 488. i)T^{2} \) |
| 29 | \( 1 + (5.16 - 52.4i)T + (-824. - 164. i)T^{2} \) |
| 31 | \( 1 + (-13.3 - 13.3i)T + 961iT^{2} \) |
| 37 | \( 1 + (19.3 + 36.2i)T + (-760. + 1.13e3i)T^{2} \) |
| 41 | \( 1 + (-16.0 + 10.7i)T + (643. - 1.55e3i)T^{2} \) |
| 43 | \( 1 + (3.22 + 2.64i)T + (360. + 1.81e3i)T^{2} \) |
| 47 | \( 1 + (-1.11 + 2.69i)T + (-1.56e3 - 1.56e3i)T^{2} \) |
| 53 | \( 1 + (-4.28 - 43.5i)T + (-2.75e3 + 548. i)T^{2} \) |
| 59 | \( 1 + (21.3 + 6.49i)T + (2.89e3 + 1.93e3i)T^{2} \) |
| 61 | \( 1 + (-50.4 + 41.3i)T + (725. - 3.64e3i)T^{2} \) |
| 67 | \( 1 + (72.5 + 88.3i)T + (-875. + 4.40e3i)T^{2} \) |
| 71 | \( 1 + (-34.6 + 6.88i)T + (4.65e3 - 1.92e3i)T^{2} \) |
| 73 | \( 1 + (41.7 + 8.30i)T + (4.92e3 + 2.03e3i)T^{2} \) |
| 79 | \( 1 + (-14.4 - 34.8i)T + (-4.41e3 + 4.41e3i)T^{2} \) |
| 83 | \( 1 + (29.0 + 15.5i)T + (3.82e3 + 5.72e3i)T^{2} \) |
| 89 | \( 1 + (42.1 - 63.0i)T + (-3.03e3 - 7.31e3i)T^{2} \) |
| 97 | \( 1 + (-55.5 + 55.5i)T - 9.40e3iT^{2} \) |
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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
−12.78047937991376019475315247597, −12.53759199957180823965633572161, −10.65896644660898227056965719039, −9.039093591728257717080692050350, −8.373696488255002213032281858984, −7.38467136222755118598038025006, −7.14158628091762845430707989377, −4.83914424997424462660312596615, −3.36259633022645096372959138381, −0.77810774371524874168999938549,
2.73431989074108222674872064209, 3.70796445184185270606667997863, 4.49583398673168817860279468572, 7.43428898319016183496534895493, 8.363377392044527737406326567451, 9.122234829149721061327111549445, 10.11361733985988316684122850363, 11.37327188320527152744126949526, 11.79353483998377558872152083224, 13.41008610301847357141856533832
