| L(s) = 1 | + 3.31i·2-s − 7·4-s − 9.38·5-s − 9.89i·7-s − 9.94i·8-s − 31.1i·10-s − 12.7i·13-s + 32.8·14-s + 5.00·16-s + 4.24i·19-s + 65.6·20-s − 28.1·23-s + 63·25-s + 42.2·26-s + 69.2i·28-s + 13.2i·29-s + ⋯ |
| L(s) = 1 | + 1.65i·2-s − 1.75·4-s − 1.87·5-s − 1.41i·7-s − 1.24i·8-s − 3.11i·10-s − 0.979i·13-s + 2.34·14-s + 0.312·16-s + 0.223i·19-s + 3.28·20-s − 1.22·23-s + 2.52·25-s + 1.62·26-s + 2.47i·28-s + 0.457i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.904 - 0.426i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5597176324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5597176324\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 3.31iT - 4T^{2} \) |
| 5 | \( 1 + 9.38T + 25T^{2} \) |
| 7 | \( 1 + 9.89iT - 49T^{2} \) |
| 13 | \( 1 + 12.7iT - 169T^{2} \) |
| 17 | \( 1 - 289T^{2} \) |
| 19 | \( 1 - 4.24iT - 361T^{2} \) |
| 23 | \( 1 + 28.1T + 529T^{2} \) |
| 29 | \( 1 - 13.2iT - 841T^{2} \) |
| 31 | \( 1 + 44T + 961T^{2} \) |
| 37 | \( 1 - 44T + 1.36e3T^{2} \) |
| 41 | \( 1 - 39.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 29.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 9.38T + 2.20e3T^{2} \) |
| 53 | \( 1 - 65.6T + 2.80e3T^{2} \) |
| 59 | \( 1 + 18.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 69.2iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 88T + 4.48e3T^{2} \) |
| 71 | \( 1 - 46.9T + 5.04e3T^{2} \) |
| 73 | \( 1 - 41.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 12.7iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 92.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 112.T + 7.92e3T^{2} \) |
| 97 | \( 1 - 70T + 9.40e3T^{2} \) |
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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
−9.886506370766587891483509298713, −8.677881388159523526484937303099, −7.895409914747986776663892539744, −7.63107037348670199349080715257, −6.99675131041694832433716493231, −5.99868530817628027886733231835, −4.81728474488088387969969853155, −4.13745357914872663926915114863, −3.41393027928313446351803696745, −0.67601336902735647994752421435,
0.29555585704730719186027147404, 1.92653863611851370656441160269, 2.86673208673362152082037337584, 3.89698079789870994027527128323, 4.37425898827163000776343733820, 5.58698683906488701723096275243, 6.97534700776859454940076125510, 7.991381119125492635960472096703, 8.822811543193578340318585871170, 9.255966152370632900748034831697
