L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (1.30 + 0.951i)5-s + (−0.809 − 0.587i)6-s + (−0.309 + 0.951i)7-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s − 1.61·10-s + (−3.04 + 1.31i)11-s + 12-s + (−2 + 1.45i)13-s + (−0.309 − 0.951i)14-s + (−0.499 + 1.53i)15-s + (−0.809 − 0.587i)16-s + (6.35 + 4.61i)17-s + ⋯ |
L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.178 + 0.549i)3-s + (0.154 − 0.475i)4-s + (0.585 + 0.425i)5-s + (−0.330 − 0.239i)6-s + (−0.116 + 0.359i)7-s + (0.109 + 0.336i)8-s + (−0.269 + 0.195i)9-s − 0.511·10-s + (−0.918 + 0.396i)11-s + 0.288·12-s + (−0.554 + 0.403i)13-s + (−0.0825 − 0.254i)14-s + (−0.129 + 0.397i)15-s + (−0.202 − 0.146i)16-s + (1.54 + 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.642 - 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.436415 + 0.935555i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.436415 + 0.935555i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.809 - 0.587i)T \) |
| 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (3.04 - 1.31i)T \) |
good | 5 | \( 1 + (-1.30 - 0.951i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (2 - 1.45i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-6.35 - 4.61i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.427 + 1.31i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 - 1.38T + 23T^{2} \) |
| 29 | \( 1 + (2.85 - 8.78i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (1.69 - 1.22i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (0.354 - 1.08i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-1.64 - 5.06i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.52T + 43T^{2} \) |
| 47 | \( 1 + (2.38 + 7.33i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-3 + 2.17i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.09 + 3.35i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.85 + 2.07i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 + 3.52T + 67T^{2} \) |
| 71 | \( 1 + (-9.70 - 7.05i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.909 - 2.80i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (-4.23 + 3.07i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.23 - 6.71i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 11.8T + 89T^{2} \) |
| 97 | \( 1 + (-10.4 + 7.60i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.97507396882410234759705872779, −10.20116888976425130228539618993, −9.738752825638085374428388087904, −8.717616448974959745495386166780, −7.82867716897433269059540987152, −6.82858564873739929109556611259, −5.73106032887712450214033376579, −4.95005608990060540620901427307, −3.29212094750592263141566266927, −2.00837673792841754386614471899,
0.74649488008124585063646568320, 2.28112067473929980417813955229, 3.39017704483368188679900497137, 5.10578476128540057792553663594, 6.01403406816116959872561187917, 7.57726226296612252423081093367, 7.75097048847614403827328340877, 9.104128597083845517417914760232, 9.783060221071561996499272595357, 10.56891826832617608713778181841