| L(s) = 1 | + 2.73·3-s + 5.26i·5-s − 4.69·7-s − 1.49·9-s + 10.2i·11-s − 17.6·13-s + 14.4i·15-s + 25.2·17-s + (−11.7 − 14.9i)19-s − 12.8·21-s − 10.4·23-s − 2.69·25-s − 28.7·27-s + 27.2·29-s − 34.2i·31-s + ⋯ |
| L(s) = 1 | + 0.913·3-s + 1.05i·5-s − 0.670·7-s − 0.166·9-s + 0.933i·11-s − 1.35·13-s + 0.961i·15-s + 1.48·17-s + (−0.618 − 0.785i)19-s − 0.612·21-s − 0.456·23-s − 0.107·25-s − 1.06·27-s + 0.940·29-s − 1.10i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.918 + 0.394i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2854679543\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2854679543\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (11.7 + 14.9i)T \) |
| good | 3 | \( 1 - 2.73T + 9T^{2} \) |
| 5 | \( 1 - 5.26iT - 25T^{2} \) |
| 7 | \( 1 + 4.69T + 49T^{2} \) |
| 11 | \( 1 - 10.2iT - 121T^{2} \) |
| 13 | \( 1 + 17.6T + 169T^{2} \) |
| 17 | \( 1 - 25.2T + 289T^{2} \) |
| 23 | \( 1 + 10.4T + 529T^{2} \) |
| 29 | \( 1 - 27.2T + 841T^{2} \) |
| 31 | \( 1 + 34.2iT - 961T^{2} \) |
| 37 | \( 1 + 69.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 45.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 7.45iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 45.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 18.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 71.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 75.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 96.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 19.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 62.7T + 5.32e3T^{2} \) |
| 79 | \( 1 - 62.4iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 101. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 73.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 107. iT - 9.40e3T^{2} \) |
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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
−10.04740252381937692887900772638, −9.290298544613709824601532738881, −8.353034193226605321069359579406, −7.37287873086918142774116424500, −7.02034964233192104464475433861, −5.93392656109076896012517664605, −4.80648957700094245926972520894, −3.58144707703573799468306265676, −2.83532565318584352041915604883, −2.13187495240462225862504198082,
0.06816786916228289692603773025, 1.52523859661290900193782967784, 2.92391489829178185134501280934, 3.51564337816468907127194378813, 4.81824667298226376249983079890, 5.58310153279431293693467175216, 6.58950845804789908604450722240, 7.78316708868528387278237736573, 8.346058107696105579951064624042, 8.938344143143239554509675620244
