| L(s) = 1 | + 4.89i·5-s + 7.74i·7-s − 12.6·11-s − 15.4i·13-s − 25.2·17-s − 8·19-s + 39.1i·23-s + 1.00·25-s + 24.4i·29-s − 7.74i·31-s − 37.9·35-s + 46.4i·37-s − 25.2·41-s + 40·43-s − 39.1i·47-s + ⋯ |
| L(s) = 1 | + 0.979i·5-s + 1.10i·7-s − 1.14·11-s − 1.19i·13-s − 1.48·17-s − 0.421·19-s + 1.70i·23-s + 0.0400·25-s + 0.844i·29-s − 0.249i·31-s − 1.08·35-s + 1.25i·37-s − 0.617·41-s + 0.930·43-s − 0.833i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.790 - 0.612i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.292924 + 0.856508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.292924 + 0.856508i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 4.89iT - 25T^{2} \) |
| 7 | \( 1 - 7.74iT - 49T^{2} \) |
| 11 | \( 1 + 12.6T + 121T^{2} \) |
| 13 | \( 1 + 15.4iT - 169T^{2} \) |
| 17 | \( 1 + 25.2T + 289T^{2} \) |
| 19 | \( 1 + 8T + 361T^{2} \) |
| 23 | \( 1 - 39.1iT - 529T^{2} \) |
| 29 | \( 1 - 24.4iT - 841T^{2} \) |
| 31 | \( 1 + 7.74iT - 961T^{2} \) |
| 37 | \( 1 - 46.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 25.2T + 1.68e3T^{2} \) |
| 43 | \( 1 - 40T + 1.84e3T^{2} \) |
| 47 | \( 1 + 39.1iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 14.6iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 25.2T + 3.48e3T^{2} \) |
| 61 | \( 1 - 15.4iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 80T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 + 10T + 5.32e3T^{2} \) |
| 79 | \( 1 - 54.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 139.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 50.5T + 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−11.86353995505031663666293740438, −10.91642565460782807113093202182, −10.29429162870582227649963198922, −9.077108608621196193332500339035, −8.116274294181486190511442436369, −7.09214519740534049322799370032, −5.95174786767460069355394983200, −5.05260582005309430837449462392, −3.23293859315296355801217254028, −2.33380271953532543606523152312,
0.42002331044309771597032873225, 2.22890814463314777838073640284, 4.22668848982062245927670690737, 4.75295136240323007128590682680, 6.32062111031044647527263642304, 7.32477418899268972819101484298, 8.444821048695255887044131700605, 9.183212665174639021596093068401, 10.43615148464997173807141804338, 11.04285550007136269455291862983
