| L(s) = 1 | − 3i·3-s − 2.82i·5-s − 14.1·7-s − 9·9-s − 20i·11-s + 39.5i·13-s − 8.48·15-s − 34·17-s + 52i·19-s + 42.4i·21-s − 62.2·23-s + 117·25-s + 27i·27-s + 200. i·29-s − 110.·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.252i·5-s − 0.763·7-s − 0.333·9-s − 0.548i·11-s + 0.844i·13-s − 0.146·15-s − 0.485·17-s + 0.627i·19-s + 0.440i·21-s − 0.564·23-s + 0.936·25-s + 0.192i·27-s + 1.28i·29-s − 0.639·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7366792226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7366792226\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3iT \) |
| good | 5 | \( 1 + 2.82iT - 125T^{2} \) |
| 7 | \( 1 + 14.1T + 343T^{2} \) |
| 11 | \( 1 + 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 39.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 34T + 4.91e3T^{2} \) |
| 19 | \( 1 - 52iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 200. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 26T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 681. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 735. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338T + 3.89e5T^{2} \) |
| 79 | \( 1 + 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
| 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
−11.21274302896840530641706007652, −10.25839028405504696884475520191, −9.125580676017375478451013815640, −8.505622132753173292373583581897, −7.23753796132891260030602296272, −6.48318552662670867163721831558, −5.49283355215879112799614352490, −4.08496260060414511643724764386, −2.83489369265530440013514325255, −1.35427127267376563876911138928,
0.25486677170884438806167587978, 2.42598263830314613612738146260, 3.55137003746182330282803491242, 4.69010627665298541652862564958, 5.85020192010675186257053136414, 6.83248560405468646779664952422, 7.899503691105174176646368595144, 9.059853747151346150785586980783, 9.804494897635004849793359926636, 10.59383195248158608611394665281
