L(s) = 1 | + (0.309 + 0.951i)2-s + (−1.72 − 0.188i)3-s + (−0.809 + 0.587i)4-s + (0.152 + 0.0496i)5-s + (−0.352 − 1.69i)6-s + (0.587 + 0.809i)7-s + (−0.809 − 0.587i)8-s + (2.92 + 0.650i)9-s + 0.160i·10-s + (3.12 + 1.11i)11-s + (1.50 − 0.859i)12-s + (−6.36 + 2.06i)13-s + (−0.587 + 0.809i)14-s + (−0.253 − 0.114i)15-s + (0.309 − 0.951i)16-s + (−0.841 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.218 + 0.672i)2-s + (−0.994 − 0.109i)3-s + (−0.404 + 0.293i)4-s + (0.0683 + 0.0221i)5-s + (−0.143 − 0.692i)6-s + (0.222 + 0.305i)7-s + (−0.286 − 0.207i)8-s + (0.976 + 0.216i)9-s + 0.0507i·10-s + (0.941 + 0.337i)11-s + (0.434 − 0.248i)12-s + (−1.76 + 0.573i)13-s + (−0.157 + 0.216i)14-s + (−0.0654 − 0.0295i)15-s + (0.0772 − 0.237i)16-s + (−0.204 + 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.908 - 0.418i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159763 + 0.728720i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159763 + 0.728720i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.309 - 0.951i)T \) |
| 3 | \( 1 + (1.72 + 0.188i)T \) |
| 7 | \( 1 + (-0.587 - 0.809i)T \) |
| 11 | \( 1 + (-3.12 - 1.11i)T \) |
good | 5 | \( 1 + (-0.152 - 0.0496i)T + (4.04 + 2.93i)T^{2} \) |
| 13 | \( 1 + (6.36 - 2.06i)T + (10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (0.841 - 2.59i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (3.45 - 4.76i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + 2.94iT - 23T^{2} \) |
| 29 | \( 1 + (5.12 - 3.72i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (-2.60 - 8.01i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (0.501 - 0.364i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (-4.44 - 3.23i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 2.41iT - 43T^{2} \) |
| 47 | \( 1 + (-1.56 + 2.15i)T + (-14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (2.46 - 0.801i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (-8.48 - 11.6i)T + (-18.2 + 56.1i)T^{2} \) |
| 61 | \( 1 + (7.68 + 2.49i)T + (49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 - 9.38T + 67T^{2} \) |
| 71 | \( 1 + (-10.0 - 3.25i)T + (57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0597 - 0.0822i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (8.72 - 2.83i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-4.39 + 13.5i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (3.20 + 9.85i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.72094469742388442412576480531, −10.47335414035735714846407545763, −9.719396593868481019902823919746, −8.649392090432689401712178211272, −7.47953532340802849880159270363, −6.70624783289717987495016100973, −5.91427979748191519830448504688, −4.84440693022425598411879899095, −4.07700521018058763763999854708, −1.93042244200188327945597409322,
0.47695820140610313237088975076, 2.24409453943415380052669054214, 3.91565948739142931837121079192, 4.81108576122957269427360213040, 5.70884016641423969837719987799, 6.85269998070229378546898492592, 7.78314285953776746842191707930, 9.502897225351791688873446853584, 9.677379589247348916609222382866, 11.02754235139102549789314855932