L(s) = 1 | − 1.26i·5-s − 7-s − 2.02i·11-s + 4.05i·13-s + 1.33·17-s − 3.28i·19-s + 2.25·23-s + 3.38·25-s + 3.58i·29-s + 0.464·31-s + 1.26i·35-s + 11.8i·37-s + 1.73·41-s − 1.70i·43-s − 8.45·47-s + ⋯ |
L(s) = 1 | − 0.567i·5-s − 0.377·7-s − 0.611i·11-s + 1.12i·13-s + 0.324·17-s − 0.753i·19-s + 0.470·23-s + 0.677·25-s + 0.666i·29-s + 0.0833·31-s + 0.214i·35-s + 1.95i·37-s + 0.271·41-s − 0.259i·43-s − 1.23·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.992 + 0.123i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.801741693\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.801741693\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 5 | \( 1 + 1.26iT - 5T^{2} \) |
| 11 | \( 1 + 2.02iT - 11T^{2} \) |
| 13 | \( 1 - 4.05iT - 13T^{2} \) |
| 17 | \( 1 - 1.33T + 17T^{2} \) |
| 19 | \( 1 + 3.28iT - 19T^{2} \) |
| 23 | \( 1 - 2.25T + 23T^{2} \) |
| 29 | \( 1 - 3.58iT - 29T^{2} \) |
| 31 | \( 1 - 0.464T + 31T^{2} \) |
| 37 | \( 1 - 11.8iT - 37T^{2} \) |
| 41 | \( 1 - 1.73T + 41T^{2} \) |
| 43 | \( 1 + 1.70iT - 43T^{2} \) |
| 47 | \( 1 + 8.45T + 47T^{2} \) |
| 53 | \( 1 + 11.7iT - 53T^{2} \) |
| 59 | \( 1 - 5.49iT - 59T^{2} \) |
| 61 | \( 1 - 6.02iT - 61T^{2} \) |
| 67 | \( 1 + 0.381iT - 67T^{2} \) |
| 71 | \( 1 - 9.47T + 71T^{2} \) |
| 73 | \( 1 + 10.0T + 73T^{2} \) |
| 79 | \( 1 + 4.90T + 79T^{2} \) |
| 83 | \( 1 + 6.95iT - 83T^{2} \) |
| 89 | \( 1 - 17.4T + 89T^{2} \) |
| 97 | \( 1 - 7.62T + 97T^{2} \) |
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
−8.267086169592225506536311491575, −7.20210518888119625541306351697, −6.69212845166960504958420793725, −6.00139913116821160706202688605, −4.99863247631526968331147004359, −4.62892875201953423689140927650, −3.53728183312394184369897662832, −2.88616762092352648319198303500, −1.71339120566926478371512380018, −0.74549912146161814047313822636,
0.67649630745103822945478978075, 1.96763904103143338714472020810, 2.90160950109082403316791182236, 3.50563469630401526207440331461, 4.42619206548088904685200549069, 5.31903053284964491769340373121, 5.98437946594622493252126960524, 6.66950807768554629592977383971, 7.49564601429359637389366537276, 7.86264039068763223801772456213