| L(s) = 1 | + 3i·3-s + 9.79i·5-s − 9.79·7-s − 9·9-s − 10i·11-s − 29.3·15-s − 29.3i·21-s − 70.9·25-s − 27i·27-s + 29.3i·29-s + 48.9·31-s + 30·33-s − 95.9i·35-s − 88.1i·45-s + 46.9·49-s + ⋯ |
| L(s) = 1 | + i·3-s + 1.95i·5-s − 1.39·7-s − 9-s − 0.909i·11-s − 1.95·15-s − 1.39i·21-s − 2.83·25-s − i·27-s + 1.01i·29-s + 1.58·31-s + 0.909·33-s − 2.74i·35-s − 1.95i·45-s + 0.959·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2898362054\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2898362054\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 - 9.79iT - 25T^{2} \) |
| 7 | \( 1 + 9.79T + 49T^{2} \) |
| 11 | \( 1 + 10iT - 121T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 + 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 29.3iT - 841T^{2} \) |
| 31 | \( 1 - 48.9T + 961T^{2} \) |
| 37 | \( 1 + 1.36e3T^{2} \) |
| 41 | \( 1 - 1.68e3T^{2} \) |
| 43 | \( 1 + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 10iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 3.72e3T^{2} \) |
| 67 | \( 1 + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 50T + 5.32e3T^{2} \) |
| 79 | \( 1 + 146.T + 6.24e3T^{2} \) |
| 83 | \( 1 + 134iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 + 190T + 9.40e3T^{2} \) |
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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
−10.60827290415047846712012902473, −10.06408642349860453590392104551, −9.368689464396729755578478336570, −8.281053992530817518595560433121, −7.06078171496695811520098921737, −6.34587557582351794192150162416, −5.71215677963127949448227127915, −4.07920436684709921228889629293, −3.11588512271652252535931808828, −2.84301004805953083384220407234,
0.10600952914724515712064259538, 1.19223801147582588829362454783, 2.47511190199634920862391305671, 3.98781826644288936700706114251, 5.04558531047826324353954237825, 5.98697049948334246615111535098, 6.80343459078781721046290642104, 7.892906539864204456003891662235, 8.551613446380586435805910516194, 9.470476573326477578139759850511
