L(s) = 1 | + (2.14 − 1.55i)2-s + (0.309 + 0.951i)3-s + (1.55 − 4.78i)4-s + (−1.38 − 1.00i)5-s + (2.14 + 1.55i)6-s + (−0.309 + 0.951i)7-s + (−2.48 − 7.64i)8-s + (−0.809 + 0.587i)9-s − 4.54·10-s + (2.49 + 2.18i)11-s + 5.03·12-s + (−1.22 + 0.888i)13-s + (0.819 + 2.52i)14-s + (0.529 − 1.62i)15-s + (−9.11 − 6.62i)16-s + (1.93 + 1.40i)17-s + ⋯ |
L(s) = 1 | + (1.51 − 1.10i)2-s + (0.178 + 0.549i)3-s + (0.777 − 2.39i)4-s + (−0.619 − 0.450i)5-s + (0.875 + 0.636i)6-s + (−0.116 + 0.359i)7-s + (−0.878 − 2.70i)8-s + (−0.269 + 0.195i)9-s − 1.43·10-s + (0.752 + 0.658i)11-s + 1.45·12-s + (−0.339 + 0.246i)13-s + (0.219 + 0.674i)14-s + (0.136 − 0.420i)15-s + (−2.27 − 1.65i)16-s + (0.469 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.94654 - 1.59178i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.94654 - 1.59178i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.309 - 0.951i)T \) |
| 7 | \( 1 + (0.309 - 0.951i)T \) |
| 11 | \( 1 + (-2.49 - 2.18i)T \) |
good | 2 | \( 1 + (-2.14 + 1.55i)T + (0.618 - 1.90i)T^{2} \) |
| 5 | \( 1 + (1.38 + 1.00i)T + (1.54 + 4.75i)T^{2} \) |
| 13 | \( 1 + (1.22 - 0.888i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (-1.93 - 1.40i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.95 - 6.02i)T + (-15.3 + 11.1i)T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + (-2.57 + 7.93i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (4.88 - 3.54i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (2.30 - 7.10i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (1.55 + 4.78i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 2.02T + 43T^{2} \) |
| 47 | \( 1 + (1.70 + 5.25i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.04 - 3.66i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-3.45 + 10.6i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (11.1 + 8.11i)T + (18.8 + 58.0i)T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + (-11.5 - 8.42i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (0.441 - 1.36i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (5.43 - 3.94i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (4.74 + 3.45i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 16.4T + 89T^{2} \) |
| 97 | \( 1 + (9.74 - 7.08i)T + (29.9 - 92.2i)T^{2} \) |
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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
−12.17232686438440665016497077842, −11.46277999507747239547154631101, −10.16787909784886881086031996824, −9.645081120849423737509484901593, −8.114324347558166895822187412431, −6.39178933921996426084360814752, −5.28436135415380082605479434694, −4.23499493401124181567715221355, −3.54886807408444071510806044479, −1.92449870108089327212593371953,
2.99936027525386509026171348800, 3.88068083516941458396028694177, 5.23135816603338273259774819082, 6.37303146741017581969942520797, 7.21980308736630114503888079461, 7.80151045782068801323053170722, 9.079899600757828469555776908658, 11.05748653068286380991531301318, 11.82198940024247396705188323542, 12.65727290090779763611751897272