| L(s) = 1 | + 6.87i·5-s + 2.64·7-s + 18.1i·11-s − 24.2·13-s − 2.43i·17-s + 0.0627·19-s − 31.9i·23-s − 22.2·25-s + 2.43i·29-s − 19.1·31-s + 18.1i·35-s + 1.12·37-s − 38.3i·41-s + 15.0·43-s + 25.0i·47-s + ⋯ |
| L(s) = 1 | + 1.37i·5-s + 0.377·7-s + 1.65i·11-s − 1.86·13-s − 0.143i·17-s + 0.00330·19-s − 1.38i·23-s − 0.889·25-s + 0.0839i·29-s − 0.616·31-s + 0.519i·35-s + 0.0304·37-s − 0.935i·41-s + 0.350·43-s + 0.533i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7910344248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7910344248\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 6.87iT - 25T^{2} \) |
| 11 | \( 1 - 18.1iT - 121T^{2} \) |
| 13 | \( 1 + 24.2T + 169T^{2} \) |
| 17 | \( 1 + 2.43iT - 289T^{2} \) |
| 19 | \( 1 - 0.0627T + 361T^{2} \) |
| 23 | \( 1 + 31.9iT - 529T^{2} \) |
| 29 | \( 1 - 2.43iT - 841T^{2} \) |
| 31 | \( 1 + 19.1T + 961T^{2} \) |
| 37 | \( 1 - 1.12T + 1.36e3T^{2} \) |
| 41 | \( 1 + 38.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 15.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 25.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 41.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 71.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 37.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 100.T + 4.48e3T^{2} \) |
| 71 | \( 1 - 66.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 36.2T + 6.24e3T^{2} \) |
| 83 | \( 1 - 156. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 135. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124.T + 9.40e3T^{2} \) |
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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
−10.36232875989379043205683307952, −10.00064218715903121603675529860, −9.006502708764450492569612584640, −7.48801802435276828877656200679, −7.33815707684699022637569978248, −6.40851868753621020101683898419, −5.05561242268921419468999443625, −4.28020800033416968573752522101, −2.77049060720728670344762583226, −2.10179627965368001254748582708,
0.26053617787314635530258894653, 1.54976854284167357739690444694, 3.05829682250630521593856315156, 4.36728518115642133539313073935, 5.22165665285636789283499477559, 5.86426347979510259132079977924, 7.33233005269799456489979454629, 8.069277953664498670859464440169, 8.905020444421467254897683166150, 9.505408694713164497839881497234
