L(s) = 1 | − 2.23·5-s + 5.25i·7-s − 19.9i·11-s + 8.47·13-s − 11.8·17-s − 15.2i·19-s − 0.555i·23-s + 5.00·25-s − 10.9·29-s + 8.29i·31-s − 11.7i·35-s + 18.3·37-s + 14.5·41-s + 22.2i·43-s + 53.3i·47-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.751i·7-s − 1.81i·11-s + 0.651·13-s − 0.699·17-s − 0.800i·19-s − 0.0241i·23-s + 0.200·25-s − 0.377·29-s + 0.267i·31-s − 0.335i·35-s + 0.496·37-s + 0.355·41-s + 0.517i·43-s + 1.13i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2070453225\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2070453225\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
good | 7 | \( 1 - 5.25iT - 49T^{2} \) |
| 11 | \( 1 + 19.9iT - 121T^{2} \) |
| 13 | \( 1 - 8.47T + 169T^{2} \) |
| 17 | \( 1 + 11.8T + 289T^{2} \) |
| 19 | \( 1 + 15.2iT - 361T^{2} \) |
| 23 | \( 1 + 0.555iT - 529T^{2} \) |
| 29 | \( 1 + 10.9T + 841T^{2} \) |
| 31 | \( 1 - 8.29iT - 961T^{2} \) |
| 37 | \( 1 - 18.3T + 1.36e3T^{2} \) |
| 41 | \( 1 - 14.5T + 1.68e3T^{2} \) |
| 43 | \( 1 - 22.2iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 53.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 66.3T + 2.80e3T^{2} \) |
| 59 | \( 1 + 17.4iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 90.1T + 3.72e3T^{2} \) |
| 67 | \( 1 - 50.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 80.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 5.55T + 5.32e3T^{2} \) |
| 79 | \( 1 - 13.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 76.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 111.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 92.8T + 9.40e3T^{2} \) |
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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
−8.303212666516631905661818370440, −7.63965481923485607270266520271, −6.47255872263297051980646381500, −6.04880904515569656643225292836, −5.17576414061591543394558355794, −4.22403418950161932487646730616, −3.27677550818976932561972609293, −2.61213511335753617377190372244, −1.19307868323845572985001694029, −0.05025849736153497714982636444,
1.35843703276170478074517095013, 2.29133310438888471140546048437, 3.61301202234096635037127331595, 4.23480782611010821193687163932, 4.89775632962409715203863573735, 6.02575903483944599660247672809, 6.86691818111703386282993466200, 7.45510852043786778566672909531, 8.054366151522921189152069651535, 9.023453080830035661483671904197