L(s) = 1 | + (−0.809 − 0.587i)2-s + (−0.309 + 0.951i)3-s + (0.309 + 0.951i)4-s + (0.309 − 0.224i)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 − 0.381·10-s + (3.04 + 1.31i)11-s − 12-s + (−1 − 0.726i)13-s + (0.309 − 0.951i)14-s + (0.118 + 0.363i)15-s + (−0.809 + 0.587i)16-s + (2.11 − 1.53i)17-s + ⋯ |
L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.178 + 0.549i)3-s + (0.154 + 0.475i)4-s + (0.138 − 0.100i)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 − 0.120·10-s + (0.918 + 0.396i)11-s − 0.288·12-s + (−0.277 − 0.201i)13-s + (0.0825 − 0.254i)14-s + (0.0304 + 0.0937i)15-s + (−0.202 + 0.146i)16-s + (0.513 − 0.373i)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
\(L(1)\) |
\(\approx\) |
\(0.915872 + 0.427234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.915872 + 0.427234i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.309 + 0.224i)T + (1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (1 + 0.726i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 1.53i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (2.42 - 7.46i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 - 3.38T + 23T^{2} \) |
| 29 | \( 1 + (0.145 + 0.449i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-3.92 - 2.85i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 4.61i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.35 - 4.16i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 1.23T + 43T^{2} \) |
| 47 | \( 1 + (-0.145 + 0.449i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.85 - 5.70i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (0.854 + 2.62i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + (-9.70 + 7.05i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (3.61 + 11.1i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (6.85 + 4.97i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-2.38 + 1.73i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 14.3T + 89T^{2} \) |
| 97 | \( 1 + (0.381 + 0.277i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.09693812997453871160839026730, −10.11477277104225010454901985303, −9.561971928335584022913248310679, −8.679922436872612618420970964012, −7.74359973530997713774106734492, −6.55120882445338372889867370042, −5.45382355835451977281309901833, −4.25496581255278465012870330881, −3.11544074081659525400107140282, −1.52806388260460076721320882733,
0.854657942454831982931819724197, 2.44340959545908893695303794126, 4.18261287068864886038838632295, 5.48657669059509500897128805703, 6.58159352263424463045867180336, 7.08470808241160050925706982838, 8.228751937387595422601025583505, 8.988448964333001409784136164352, 9.947081104368732497996672958015, 10.96595307576124945593833762686