| L(s) = 1 | − 724. i·2-s − 5.24e5·4-s − 1.79e7i·5-s + 3.70e8·7-s + 3.79e8i·8-s − 1.29e10·10-s + 7.93e9i·11-s + 8.82e10·13-s − 2.68e11i·14-s + 2.74e11·16-s + 1.27e12i·17-s + 6.13e11·19-s + 9.39e12i·20-s + 5.74e12·22-s − 5.25e13i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s − 1.83i·5-s + 1.31·7-s + 0.353i·8-s − 1.29·10-s + 0.306i·11-s + 0.639·13-s − 0.926i·14-s + 0.250·16-s + 0.634i·17-s + 0.100·19-s + 0.917i·20-s + 0.216·22-s − 1.26i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(21-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+10) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{21}{2})\) |
\(\approx\) |
\(2.401097579\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.401097579\) |
| \(L(11)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 724. iT \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.79e7iT - 9.53e13T^{2} \) |
| 7 | \( 1 - 3.70e8T + 7.97e16T^{2} \) |
| 11 | \( 1 - 7.93e9iT - 6.72e20T^{2} \) |
| 13 | \( 1 - 8.82e10T + 1.90e22T^{2} \) |
| 17 | \( 1 - 1.27e12iT - 4.06e24T^{2} \) |
| 19 | \( 1 - 6.13e11T + 3.75e25T^{2} \) |
| 23 | \( 1 + 5.25e13iT - 1.71e27T^{2} \) |
| 29 | \( 1 + 4.94e14iT - 1.76e29T^{2} \) |
| 31 | \( 1 - 4.08e14T + 6.71e29T^{2} \) |
| 37 | \( 1 - 8.86e15T + 2.31e31T^{2} \) |
| 41 | \( 1 + 1.66e16iT - 1.80e32T^{2} \) |
| 43 | \( 1 + 5.50e15T + 4.67e32T^{2} \) |
| 47 | \( 1 - 6.38e16iT - 2.76e33T^{2} \) |
| 53 | \( 1 + 2.19e17iT - 3.05e34T^{2} \) |
| 59 | \( 1 - 6.16e17iT - 2.61e35T^{2} \) |
| 61 | \( 1 - 4.13e17T + 5.08e35T^{2} \) |
| 67 | \( 1 - 7.34e17T + 3.32e36T^{2} \) |
| 71 | \( 1 + 5.12e18iT - 1.05e37T^{2} \) |
| 73 | \( 1 - 5.57e18T + 1.84e37T^{2} \) |
| 79 | \( 1 + 9.30e18T + 8.96e37T^{2} \) |
| 83 | \( 1 + 1.56e19iT - 2.40e38T^{2} \) |
| 89 | \( 1 - 2.32e19iT - 9.72e38T^{2} \) |
| 97 | \( 1 + 1.26e20T + 5.43e39T^{2} \) |
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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
−11.01752035202128307926601201109, −9.649486541505881567599367848709, −8.521293896271341765223635474401, −8.042537343410528221059353439634, −5.80970798164015739977617320280, −4.67715866876792328456508761855, −4.16805259257258993225843481508, −2.17046842233906367165552921885, −1.28165367458588657318740540477, −0.52518996046865121434957378742,
1.25619545860242369568464157011, 2.68117129886177483139840419365, 3.80655377056121536276849765630, 5.25288855005787435363980336626, 6.39481919520815443692258155063, 7.36675059463206705409901443427, 8.202691379162525951692361311066, 9.726769849222503049435259665557, 10.99996714192682886291991186280, 11.50903042092732907088069967989
