L(s) = 1 | + i·2-s + (−0.167 − 1.72i)3-s − 4-s + (−1.17 + 2.03i)5-s + (1.72 − 0.167i)6-s − i·8-s + (−2.94 + 0.577i)9-s + (−2.03 − 1.17i)10-s + (4.91 − 2.83i)11-s + (0.167 + 1.72i)12-s + (−1.48 + 0.859i)13-s + (3.70 + 1.68i)15-s + 16-s + (−0.884 + 1.53i)17-s + (−0.577 − 2.94i)18-s + (0.986 − 0.569i)19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.0967 − 0.995i)3-s − 0.5·4-s + (−0.525 + 0.909i)5-s + (0.703 − 0.0684i)6-s − 0.353i·8-s + (−0.981 + 0.192i)9-s + (−0.643 − 0.371i)10-s + (1.48 − 0.855i)11-s + (0.0483 + 0.497i)12-s + (−0.413 + 0.238i)13-s + (0.956 + 0.434i)15-s + 0.250·16-s + (−0.214 + 0.371i)17-s + (−0.136 − 0.693i)18-s + (0.226 − 0.130i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10397 + 0.605331i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10397 + 0.605331i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.167 + 1.72i)T \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.17 - 2.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.91 + 2.83i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.48 - 0.859i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (0.884 - 1.53i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.986 + 0.569i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.18 - 1.83i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.59 - 2.07i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.37iT - 31T^{2} \) |
| 37 | \( 1 + (-4.59 - 7.96i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.99 + 6.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.76 + 3.04i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 11.8T + 47T^{2} \) |
| 53 | \( 1 + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 - 2.22T + 59T^{2} \) |
| 61 | \( 1 - 8.99iT - 61T^{2} \) |
| 67 | \( 1 - 10.8T + 67T^{2} \) |
| 71 | \( 1 - 4.52iT - 71T^{2} \) |
| 73 | \( 1 + (-4.62 - 2.67i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + 13.0T + 79T^{2} \) |
| 83 | \( 1 + (-6.27 + 10.8i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (0.580 + 1.00i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.97 - 2.29i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.35625084084367117691419409153, −8.980805834038878321469862722119, −8.560429917509079125768482391384, −7.39751014680900448218897434858, −6.88465506943277761351184379551, −6.32047611321033741386595597202, −5.26418197300941647936529461397, −3.85345593274844880723703852469, −2.90568335653686949681791251208, −1.16321884396700978557865779389,
0.77088162248007306029638269857, 2.50251104155536634637322391672, 3.90795099820617408489073645146, 4.38941831565909921545600840522, 5.17574877087187367609162921382, 6.40128780162699181851720108998, 7.72462096290600754253514724471, 8.679006475633168748969772031491, 9.411103785644058028805546364740, 9.780509729073915971573100157302