L(s) = 1 | + (−0.918 − 0.667i)2-s + (−0.219 − 0.675i)4-s + (0.809 − 0.587i)5-s + (1.26 + 3.90i)7-s + (−0.951 + 2.92i)8-s − 1.13·10-s + (−3.26 + 0.600i)11-s + (1.83 + 1.33i)13-s + (1.43 − 4.42i)14-s + (1.67 − 1.21i)16-s + (−4.06 + 2.95i)17-s + (−2.34 + 7.20i)19-s + (−0.574 − 0.417i)20-s + (3.39 + 1.62i)22-s + 1.97·23-s + ⋯ |
L(s) = 1 | + (−0.649 − 0.472i)2-s + (−0.109 − 0.337i)4-s + (0.361 − 0.262i)5-s + (0.479 + 1.47i)7-s + (−0.336 + 1.03i)8-s − 0.359·10-s + (−0.983 + 0.180i)11-s + (0.510 + 0.370i)13-s + (0.384 − 1.18i)14-s + (0.419 − 0.304i)16-s + (−0.984 + 0.715i)17-s + (−0.537 + 1.65i)19-s + (−0.128 − 0.0933i)20-s + (0.724 + 0.346i)22-s + 0.411·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.739812 + 0.327738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.739812 + 0.327738i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (3.26 - 0.600i)T \) |
good | 2 | \( 1 + (0.918 + 0.667i)T + (0.618 + 1.90i)T^{2} \) |
| 7 | \( 1 + (-1.26 - 3.90i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (-1.83 - 1.33i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (4.06 - 2.95i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.34 - 7.20i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 1.97T + 23T^{2} \) |
| 29 | \( 1 + (1.23 + 3.80i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.08 - 2.24i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.07 + 6.40i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.24 - 3.82i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 9.57T + 43T^{2} \) |
| 47 | \( 1 + (0.984 - 3.03i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-9.08 - 6.60i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.94 - 9.07i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (2.92 - 2.12i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + 6.31T + 67T^{2} \) |
| 71 | \( 1 + (-9.94 + 7.22i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (1.20 + 3.69i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (14.2 + 10.3i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-11.2 + 8.13i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 7.85T + 89T^{2} \) |
| 97 | \( 1 + (4.18 + 3.04i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.83435386495423445820333703638, −10.26767332589362710140582155812, −9.133295207444182690421187789972, −8.691872908465956170853479170470, −7.88528423981391857679656195297, −6.05614761843357483558136206554, −5.64559192381097902841024127588, −4.46425242257326435279930543940, −2.50188789448815569733347204218, −1.72020141475238509847768450202,
0.60501211488794861593195538275, 2.75155728635895270603486180806, 4.06613993418274952146505226905, 5.11000008096658023019984207386, 6.74269248116938797103554611373, 7.12868232787213810365713302220, 8.124721197851996512121149398155, 8.871366808685221409600153130573, 9.911729259576755474152237274354, 10.76613889119379507593102122605