L(s) = 1 | + (2.86 − 22.4i)2-s + 247. i·3-s + (−495. − 128. i)4-s + 1.41e3i·5-s + (5.55e3 + 709. i)6-s + 5.08e3·7-s + (−4.31e3 + 1.07e4i)8-s − 4.15e4·9-s + (3.18e4 + 4.06e3i)10-s − 1.48e4i·11-s + (3.18e4 − 1.22e5i)12-s + 6.40e4i·13-s + (1.45e4 − 1.14e5i)14-s − 3.50e5·15-s + (2.28e5 + 1.27e5i)16-s + 2.51e5·17-s + ⋯ |
L(s) = 1 | + (0.126 − 0.991i)2-s + 1.76i·3-s + (−0.967 − 0.251i)4-s + 1.01i·5-s + (1.74 + 0.223i)6-s + 0.800·7-s + (−0.372 + 0.928i)8-s − 2.10·9-s + (1.00 + 0.128i)10-s − 0.305i·11-s + (0.443 − 1.70i)12-s + 0.622i·13-s + (0.101 − 0.794i)14-s − 1.78·15-s + (0.873 + 0.486i)16-s + 0.729·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.372 - 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(1.14076 + 0.771520i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.14076 + 0.771520i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-2.86 + 22.4i)T \) |
good | 3 | \( 1 - 247. iT - 1.96e4T^{2} \) |
| 5 | \( 1 - 1.41e3iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 5.08e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.48e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 6.40e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 2.51e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.11e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 1.96e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.16e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 + 3.03e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 8.74e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 1.48e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.53e5iT - 5.02e14T^{2} \) |
| 47 | \( 1 - 5.17e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 4.26e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 1.15e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 + 1.95e8iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.69e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 - 2.23e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.10e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.51e7T + 1.19e17T^{2} \) |
| 83 | \( 1 + 4.73e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 - 8.45e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.97e8T + 7.60e17T^{2} \) |
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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
−20.40902333459441252962047641186, −18.75742403191153948371137546485, −17.07247912318096439514229388370, −15.10227852350045617971139098218, −14.19248293819079597878766064704, −11.34888803696935809559397074830, −10.57606590784860215865387190350, −9.066119657328896677898637071310, −4.91200434140504340858827066423, −3.19765859938353322157558525430,
1.02553271768728285703819471725, 5.47507876430754981413015897677, 7.42601265088147744082964880155, 8.539255450859165808688795095002, 12.26000126045001234446110840509, 13.20262344988885223296115608556, 14.62244718858995228697031701352, 16.81593882016075729106360703200, 17.78133996042136960815267228662, 18.96007466625238273116851262794