| L(s) = 1 | − 4i·7-s − 4i·11-s + 2·13-s + (−1 + 4i)17-s + 4·19-s + 4i·23-s + 5·25-s − 4i·31-s − 8i·37-s − 8i·41-s + 4·43-s − 8·47-s − 9·49-s − 6·53-s − 12·59-s + ⋯ |
| L(s) = 1 | − 1.51i·7-s − 1.20i·11-s + 0.554·13-s + (−0.242 + 0.970i)17-s + 0.917·19-s + 0.834i·23-s + 25-s − 0.718i·31-s − 1.31i·37-s − 1.24i·41-s + 0.609·43-s − 1.16·47-s − 1.28·49-s − 0.824·53-s − 1.56·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.607872956\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.607872956\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 17 | \( 1 + (1 - 4i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 + 4iT - 7T^{2} \) |
| 11 | \( 1 + 4iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + 8iT - 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 12T + 59T^{2} \) |
| 61 | \( 1 - 8iT - 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.754876984127810087531641499762, −7.78987309809769759559479969485, −7.36375332958445321348303956078, −6.33860563095576610389997167655, −5.73637868312127189236441813779, −4.62365877530828677142891695210, −3.73200549467605985910475138340, −3.21488755984848636972305576028, −1.57571465254573096698888981131, −0.57352570508392388865017562551,
1.41278891630442857583352756514, 2.55317958112811488980896090028, 3.19891141047355361785014440707, 4.78669127463215302246567304886, 4.96710679513710451000328063963, 6.16631158193617952287891358636, 6.73094065576395437782977227829, 7.70263282682119755145400937233, 8.478535596709911215007179621237, 9.225287709285959191736619465834
