| L(s) = 1 | + (2.82 − i)3-s − 6i·7-s + (7.00 − 5.65i)9-s + 5.65i·11-s + 10i·13-s + 22.6·17-s + 2·19-s + (−6 − 16.9i)21-s + 11.3·23-s + (14.1 − 23.0i)27-s + 16.9i·29-s + 22·31-s + (5.65 + 16.0i)33-s + 6i·37-s + (10 + 28.2i)39-s + ⋯ |
| L(s) = 1 | + (0.942 − 0.333i)3-s − 0.857i·7-s + (0.777 − 0.628i)9-s + 0.514i·11-s + 0.769i·13-s + 1.33·17-s + 0.105·19-s + (−0.285 − 0.808i)21-s + 0.491·23-s + (0.523 − 0.851i)27-s + 0.585i·29-s + 0.709·31-s + (0.171 + 0.484i)33-s + 0.162i·37-s + (0.256 + 0.725i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.719 + 0.694i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.016654461\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.016654461\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.82 + i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 6iT - 49T^{2} \) |
| 11 | \( 1 - 5.65iT - 121T^{2} \) |
| 13 | \( 1 - 10iT - 169T^{2} \) |
| 17 | \( 1 - 22.6T + 289T^{2} \) |
| 19 | \( 1 - 2T + 361T^{2} \) |
| 23 | \( 1 - 11.3T + 529T^{2} \) |
| 29 | \( 1 - 16.9iT - 841T^{2} \) |
| 31 | \( 1 - 22T + 961T^{2} \) |
| 37 | \( 1 - 6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 33.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 82iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 67.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 62.2T + 2.80e3T^{2} \) |
| 59 | \( 1 + 73.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 86T + 3.72e3T^{2} \) |
| 67 | \( 1 - 2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 124. iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 82iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 10T + 6.24e3T^{2} \) |
| 83 | \( 1 - 73.5T + 6.88e3T^{2} \) |
| 89 | \( 1 + 33.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 94iT - 9.40e3T^{2} \) |
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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
−9.391315447326588680859090236986, −8.678861478107384049022290999875, −7.63402198411879249270306355354, −7.24978780252556409182388607985, −6.37114532820351870517863009458, −5.03814207501648209222907801610, −4.03068789230044918146839830424, −3.27662824567679045423743814768, −2.02869926988907983939862029892, −0.947275998151289551516037014993,
1.20575324957054945883953449563, 2.69425762212409469461542112496, 3.18881363189275794261098118281, 4.40779874694808287284868531541, 5.41632925145245739796873318481, 6.18654984821441624099324595274, 7.55149664404458741229586110050, 8.054436853776840696576780057665, 8.848638624078785071083263181555, 9.575181835008413321440826966039
