| L(s) = 1 | − 3-s − 5.19i·5-s − 8·9-s − 17·11-s − 13.8i·13-s + 5.19i·15-s − 25·17-s + 7·19-s − 5.19i·23-s − 2·25-s + 17·27-s + 13.8i·29-s + 32.9i·31-s + 17·33-s + 8.66i·37-s + ⋯ |
| L(s) = 1 | − 0.333·3-s − 1.03i·5-s − 0.888·9-s − 1.54·11-s − 1.06i·13-s + 0.346i·15-s − 1.47·17-s + 0.368·19-s − 0.225i·23-s − 0.0800·25-s + 0.629·27-s + 0.477i·29-s + 1.06i·31-s + 0.515·33-s + 0.234i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1568 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4895897940\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4895897940\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + 9T^{2} \) |
| 5 | \( 1 + 5.19iT - 25T^{2} \) |
| 11 | \( 1 + 17T + 121T^{2} \) |
| 13 | \( 1 + 13.8iT - 169T^{2} \) |
| 17 | \( 1 + 25T + 289T^{2} \) |
| 19 | \( 1 - 7T + 361T^{2} \) |
| 23 | \( 1 + 5.19iT - 529T^{2} \) |
| 29 | \( 1 - 13.8iT - 841T^{2} \) |
| 31 | \( 1 - 32.9iT - 961T^{2} \) |
| 37 | \( 1 - 8.66iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 26T + 1.68e3T^{2} \) |
| 43 | \( 1 + 14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 50.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 91.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 55T + 3.48e3T^{2} \) |
| 61 | \( 1 - 22.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 17T + 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 119T + 5.32e3T^{2} \) |
| 79 | \( 1 + 74.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 110T + 6.88e3T^{2} \) |
| 89 | \( 1 - 71T + 7.92e3T^{2} \) |
| 97 | \( 1 + 22T + 9.40e3T^{2} \) |
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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
−9.226681735418963901737512452034, −8.464790529258025557998379118619, −8.068441847038769378233413373198, −6.96436335428528920012894623192, −5.89977193404019294035735033885, −5.14762795094353522404674793220, −4.75428663392179419049555187533, −3.25190631379437333822161851679, −2.34922867621252734167130544817, −0.77185455613480110012205615524,
0.18758579807802497135973870590, 2.24911892018452071153316346585, 2.78071235242459858254878995313, 4.03126787666172070888545901443, 5.05821808081289894260915059764, 5.91974074484737001264117780466, 6.67402285489190468793034546011, 7.40721291082292431501327823937, 8.279066911135809340545206980184, 9.123898892407544240494017593281
