L(s) = 1 | + (−0.809 − 0.587i)2-s + (1.61 + 0.622i)3-s + (0.309 + 0.951i)4-s + (−1.77 − 2.44i)5-s + (−0.941 − 1.45i)6-s + (−0.951 + 0.309i)7-s + (0.309 − 0.951i)8-s + (2.22 + 2.01i)9-s + 3.02i·10-s + (2.94 − 1.51i)11-s + (−0.0930 + 1.72i)12-s + (1.46 − 2.01i)13-s + (0.951 + 0.309i)14-s + (−1.34 − 5.05i)15-s + (−0.809 + 0.587i)16-s + (2.63 − 1.91i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (0.933 + 0.359i)3-s + (0.154 + 0.475i)4-s + (−0.794 − 1.09i)5-s + (−0.384 − 0.593i)6-s + (−0.359 + 0.116i)7-s + (0.109 − 0.336i)8-s + (0.741 + 0.671i)9-s + 0.955i·10-s + (0.889 − 0.457i)11-s + (−0.0268 + 0.499i)12-s + (0.406 − 0.559i)13-s + (0.254 + 0.0825i)14-s + (−0.347 − 1.30i)15-s + (−0.202 + 0.146i)16-s + (0.640 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.344 + 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.04408 - 0.728639i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.04408 - 0.728639i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.61 - 0.622i)T \) |
| 7 | \( 1 + (0.951 - 0.309i)T \) |
| 11 | \( 1 + (-2.94 + 1.51i)T \) |
good | 5 | \( 1 + (1.77 + 2.44i)T + (-1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.46 + 2.01i)T + (-4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.63 + 1.91i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.06 + 0.671i)T + (15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 7.69iT - 23T^{2} \) |
| 29 | \( 1 + (2.83 + 8.71i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.14 - 3.73i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.36 - 4.21i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (3.00 - 9.25i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + 7.49iT - 43T^{2} \) |
| 47 | \( 1 + (-9.50 - 3.08i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.92 - 8.15i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (4.10 - 1.33i)T + (47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (1.82 + 2.51i)T + (-18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + (6.23 + 8.57i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.37 - 1.74i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (1.33 - 1.83i)T + (-24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (7.58 - 5.51i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 - 11.7iT - 89T^{2} \) |
| 97 | \( 1 + (-10.8 - 7.87i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.71541103839368952405112033021, −9.797391336994483977826563377229, −8.952059581768980149947841570388, −8.395426312866232479632699641845, −7.74182176808444088953642316055, −6.35366079354053361308216482842, −4.68163818626694219423475133536, −3.86809433294082325453181863354, −2.75427289512853275067686047280, −0.953720701274878392070113158461,
1.68257812746593136614925838897, 3.28005776784754513979335721991, 4.03717036282022206316087304116, 6.04620461967057152314612127539, 7.06576101094258605889482919321, 7.36933944574107959175898999582, 8.459921550842876879044145537593, 9.312164949449737581066146917673, 10.11344523330064521233162465369, 11.13614903376590170150119351665