| L(s) = 1 | + 15.5·3-s − 20i·5-s + 529. i·7-s + 243·9-s + 435.·11-s + 341. i·13-s − 311. i·15-s + 7.68e3·17-s − 4.30e3·19-s + 8.25e3i·21-s − 3.17e3i·23-s + 1.52e4·25-s + 3.78e3·27-s − 1.94e4i·29-s + 1.52e4i·31-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.160i·5-s + 1.54i·7-s + 0.333·9-s + 0.327·11-s + 0.155i·13-s − 0.0923i·15-s + 1.56·17-s − 0.626·19-s + 0.891i·21-s − 0.260i·23-s + 0.974·25-s + 0.192·27-s − 0.795i·29-s + 0.513i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.613303952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.613303952\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 15.5T \) |
| good | 5 | \( 1 + 20iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 529. iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 435.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 341. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 7.68e3T + 2.41e7T^{2} \) |
| 19 | \( 1 + 4.30e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 3.17e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.94e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 - 1.52e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 6.19e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 3.37e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 9.93e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.77e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 2.24e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 1.99e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 4.56e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 + 4.96e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.52e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.94e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 5.71e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 3.24e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 7.58e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 2.50e4T + 8.32e11T^{2} \) |
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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
−10.37475009807122710433338073589, −9.501718782530458718029127619327, −8.668428576396202020634971631638, −8.089257996738334909749628182349, −6.75632322525468159651359487844, −5.75043372055172399024420265517, −4.77015767518912019998477557845, −3.36504295382554983447618911512, −2.44132941208016368323196081874, −1.26103648517830252407875531868,
0.56523714785408119841472428900, 1.62077901557472089359438931819, 3.21890734905149019689509879670, 3.91506123677362789707811725441, 5.10493363208262211575299654447, 6.55574404013286361219000797159, 7.37013048196495235515487507469, 8.107538850291593506510739822745, 9.242199767428124816642404893397, 10.22811087655399796309885771055
