L(s) = 1 | + (−0.866 − 1.5i)2-s + (−0.5 + 0.866i)4-s + (1.73 + 3i)5-s − 1.73·8-s + (3 − 5.19i)10-s + (−1.73 + 3i)11-s + 2·13-s + (2.49 + 4.33i)16-s + (−1.73 + 3i)17-s + (2 + 3.46i)19-s − 3.46·20-s + 6·22-s + (1.73 + 3i)23-s + (−3.5 + 6.06i)25-s + (−1.73 − 3i)26-s + ⋯ |
L(s) = 1 | + (−0.612 − 1.06i)2-s + (−0.250 + 0.433i)4-s + (0.774 + 1.34i)5-s − 0.612·8-s + (0.948 − 1.64i)10-s + (−0.522 + 0.904i)11-s + 0.554·13-s + (0.624 + 1.08i)16-s + (−0.420 + 0.727i)17-s + (0.458 + 0.794i)19-s − 0.774·20-s + 1.27·22-s + (0.361 + 0.625i)23-s + (−0.700 + 1.21i)25-s + (−0.339 − 0.588i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04087 + 0.0660541i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04087 + 0.0660541i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 1.5i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (-1.73 - 3i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.73 - 3i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (1.73 - 3i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 3i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 10.3T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (3.46 + 6i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-3.46 + 6i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.46 + 6i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 - 8.66i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 + 3.46i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + (7 - 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4 + 6.92i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-1.73 - 3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−10.98887805739241215794199083103, −10.11947764587308644532347165234, −9.894407666243160427149782490803, −8.727011964416920478322256094674, −7.50958530219792359034937777842, −6.45518653816995207796827242672, −5.61293143883103867370667363314, −3.76463270586229244986735471166, −2.64925095498153368402024617772, −1.74670460510648390999590675763,
0.828840513664867935599289602033, 2.85636413025418105774924826161, 4.73923974954126445975856877768, 5.59467185202947558000675716199, 6.40141010493233504366606581153, 7.53241019937762652152513424655, 8.602083726209326506024976057020, 8.901289151620908951317373383751, 9.768988302378158257617624412363, 11.01032789922896324300908660310