L(s) = 1 | − 5.23i·3-s − 2.23·5-s + 10.1i·7-s − 18.4·9-s + 14.4i·11-s − 11.5·13-s + 11.7i·15-s − 18.9·17-s − 12i·19-s + 53.3·21-s + 17.5i·23-s + 5.00·25-s + 49.3i·27-s − 8.83·29-s − 0.583i·31-s + ⋯ |
L(s) = 1 | − 1.74i·3-s − 0.447·5-s + 1.45i·7-s − 2.04·9-s + 1.31i·11-s − 0.886·13-s + 0.780i·15-s − 1.11·17-s − 0.631i·19-s + 2.53·21-s + 0.765i·23-s + 0.200·25-s + 1.82i·27-s − 0.304·29-s − 0.0188i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.301416 + 0.301416i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.301416 + 0.301416i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23T \) |
good | 3 | \( 1 + 5.23iT - 9T^{2} \) |
| 7 | \( 1 - 10.1iT - 49T^{2} \) |
| 11 | \( 1 - 14.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.5T + 169T^{2} \) |
| 17 | \( 1 + 18.9T + 289T^{2} \) |
| 19 | \( 1 + 12iT - 361T^{2} \) |
| 23 | \( 1 - 17.5iT - 529T^{2} \) |
| 29 | \( 1 + 8.83T + 841T^{2} \) |
| 31 | \( 1 + 0.583iT - 961T^{2} \) |
| 37 | \( 1 + 32.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 71.3T + 1.68e3T^{2} \) |
| 43 | \( 1 - 4.65iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 22.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 63.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 30.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 65.1T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 41.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 136.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 81.1iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 86.5iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 30T + 7.92e3T^{2} \) |
| 97 | \( 1 - 119.T + 9.40e3T^{2} \) |
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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
−11.96254670967363271466674859370, −11.17434759486201771067959819568, −9.471923282046468782314846167831, −8.687101469936108377824594651527, −7.59513273535892662551633056633, −7.01857639029786317974522175149, −5.97105651372642004445965981872, −4.80475485804134844693545719976, −2.66542043612156807233549836701, −1.88887330214838496435195585963,
0.19023387039551580681263350849, 3.12154046170185216552770551822, 4.08754090670782577984927154659, 4.72424217989125267584584158657, 6.08149080806083374724502048236, 7.46991837573269086921773041072, 8.562337312802047569699047714583, 9.434147103885141553749352098537, 10.53790049894665199303736330498, 10.76582697814676144058860546168