| L(s) = 1 | + (−0.978 + 0.207i)2-s + (0.104 + 0.994i)3-s + (0.913 − 0.406i)4-s + (−2.35 + 2.62i)5-s + (−0.309 − 0.951i)6-s + (0.972 − 2.46i)7-s + (−0.809 + 0.587i)8-s + (−0.978 + 0.207i)9-s + (1.76 − 3.05i)10-s + (−2.91 − 1.57i)11-s + (0.5 + 0.866i)12-s + (−1.54 + 4.74i)13-s + (−0.439 + 2.60i)14-s + (−2.85 − 2.07i)15-s + (0.669 − 0.743i)16-s + (−3.15 − 0.669i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 + 0.147i)2-s + (0.0603 + 0.574i)3-s + (0.456 − 0.203i)4-s + (−1.05 + 1.17i)5-s + (−0.126 − 0.388i)6-s + (0.367 − 0.930i)7-s + (−0.286 + 0.207i)8-s + (−0.326 + 0.0693i)9-s + (0.557 − 0.965i)10-s + (−0.880 − 0.474i)11-s + (0.144 + 0.250i)12-s + (−0.427 + 1.31i)13-s + (−0.117 + 0.697i)14-s + (−0.736 − 0.535i)15-s + (0.167 − 0.185i)16-s + (−0.764 − 0.162i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.775 + 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0351017 - 0.0987800i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0351017 - 0.0987800i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 - 0.207i)T \) |
| 3 | \( 1 + (-0.104 - 0.994i)T \) |
| 7 | \( 1 + (-0.972 + 2.46i)T \) |
| 11 | \( 1 + (2.91 + 1.57i)T \) |
| good | 5 | \( 1 + (2.35 - 2.62i)T + (-0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (1.54 - 4.74i)T + (-10.5 - 7.64i)T^{2} \) |
| 17 | \( 1 + (3.15 + 0.669i)T + (15.5 + 6.91i)T^{2} \) |
| 19 | \( 1 + (-2.13 - 0.948i)T + (12.7 + 14.1i)T^{2} \) |
| 23 | \( 1 + (4.26 + 7.38i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.70 - 2.69i)T + (8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (6.21 + 6.90i)T + (-3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-0.0234 + 0.223i)T + (-36.1 - 7.69i)T^{2} \) |
| 41 | \( 1 + (3.16 - 2.29i)T + (12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + 9.81T + 43T^{2} \) |
| 47 | \( 1 + (-8.60 - 3.82i)T + (31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (2.08 + 2.32i)T + (-5.54 + 52.7i)T^{2} \) |
| 59 | \( 1 + (0.654 - 0.291i)T + (39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (4.17 - 4.63i)T + (-6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (4.70 - 8.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.645 - 1.98i)T + (-57.4 + 41.7i)T^{2} \) |
| 73 | \( 1 + (8.24 - 3.66i)T + (48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.42 + 1.36i)T + (72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.33 + 7.19i)T + (-67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-6.65 - 11.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.16 + 6.66i)T + (-78.4 - 57.0i)T^{2} \) |
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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
−11.29026829288079969828648186979, −10.63110471741345052324756950461, −10.06532961976334005080404464178, −8.816019224586942607646856099885, −7.892216809139868931619125814968, −7.21219127037718840357462655000, −6.36914854252173479470901832076, −4.68428912420807627844362387755, −3.76143376549941484514018619097, −2.47590895516629664587286937789,
0.07626696104874750096041790350, 1.78929532014864308155215349647, 3.21199287409201404871892526449, 4.87112932974380647933738986856, 5.64016355382060256414241066719, 7.28934162017007543252405319933, 7.959945334280716725967552798371, 8.497364947186116094638190483859, 9.376358220603912055736116792878, 10.52592490130463345822844259477
