| L(s) = 1 | − 1.73i·3-s + 2.82·5-s − 4.89i·7-s − 2.99·9-s + 3.46i·11-s − 4.89i·15-s − 8.48·21-s + 3.00·25-s + 5.19i·27-s + 2.82·29-s − 4.89i·31-s + 5.99·33-s − 13.8i·35-s − 8.48·45-s − 16.9·49-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 1.26·5-s − 1.85i·7-s − 0.999·9-s + 1.04i·11-s − 1.26i·15-s − 1.85·21-s + 0.600·25-s + 0.999i·27-s + 0.525·29-s − 0.879i·31-s + 1.04·33-s − 2.34i·35-s − 1.26·45-s − 2.42·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.10870 - 1.10870i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.10870 - 1.10870i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 7 | \( 1 + 4.89iT - 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 4.89iT - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 10.3iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 - 14.6iT - 79T^{2} \) |
| 83 | \( 1 - 17.3iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−10.99610042436231843172565510573, −10.17667333838459606769819445249, −9.473631002461414628881229428724, −8.093540629927526297741136445291, −7.16049473788098287931266862280, −6.60591535003832294623708602413, −5.42512073171419559766799484655, −4.09027501861879344834821570027, −2.36969536973786420608727183049, −1.14793255303810996878577054749,
2.24922053309418090877101395844, 3.24064585936274133936720393286, 5.01786783295086349677149415785, 5.69497222128689072308810440741, 6.31888510608475523802588580068, 8.387139524508073143740744199665, 8.932902022988672480944492989716, 9.641506657524194233949913949317, 10.52407573646537156370235543594, 11.48983946172269775987873204758
