L(s) = 1 | − 1.41i·2-s + 2.44·5-s − 2.82i·8-s − 3.46i·10-s − 5.65i·11-s − 3.46i·13-s − 4.00·16-s + 2.44·17-s + 1.73i·19-s − 8.00·22-s + 7.07i·23-s + 0.999·25-s − 4.89·26-s − 1.41i·29-s + 5.19i·31-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + 1.09·5-s − 0.999i·8-s − 1.09i·10-s − 1.70i·11-s − 0.960i·13-s − 1.00·16-s + 0.594·17-s + 0.397i·19-s − 1.70·22-s + 1.47i·23-s + 0.199·25-s − 0.960·26-s − 0.262i·29-s + 0.933i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.654 + 0.755i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.214549617\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.214549617\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.41iT - 2T^{2} \) |
| 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 5.65iT - 11T^{2} \) |
| 13 | \( 1 + 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 2.44T + 17T^{2} \) |
| 19 | \( 1 - 1.73iT - 19T^{2} \) |
| 23 | \( 1 - 7.07iT - 23T^{2} \) |
| 29 | \( 1 + 1.41iT - 29T^{2} \) |
| 31 | \( 1 - 5.19iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 + 7T + 43T^{2} \) |
| 47 | \( 1 - 7.34T + 47T^{2} \) |
| 53 | \( 1 + 1.41iT - 53T^{2} \) |
| 59 | \( 1 - 14.6T + 59T^{2} \) |
| 61 | \( 1 + 5.19iT - 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 + 12.1iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 9.79T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 8.66iT - 97T^{2} \) |
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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
−9.646142685728487693478572222059, −8.737027216024998708475098370479, −7.85032912549529423288447586391, −6.73160133576794666526167745560, −5.81106970216465401383198068022, −5.34762215962010658576289169963, −3.58292376666547518646566400215, −3.14775754718984525975416580052, −1.94569743290888129235309167133, −0.915507227025875453357793723236,
1.83388915671197582229556858611, 2.46993964789137036973202544275, 4.27648783720070339870586111494, 5.09697202808740114855547135786, 5.90061519650483049965513452327, 6.83319493251373109573172059498, 7.09546934070864940502421992157, 8.220897700791043483664701886698, 9.054497509279055628752783890422, 9.845864557982950823005236936671