| L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.168 + 1.60i)5-s + (0.809 + 0.587i)6-s + (−1.86 − 1.87i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (0.806 − 1.39i)10-s + (3.31 + 0.0483i)11-s + (−0.499 − 0.866i)12-s + (0.214 − 0.156i)13-s + (0.945 + 2.47i)14-s + (0.498 − 1.53i)15-s + (−0.104 + 0.994i)16-s + (−4.96 + 2.21i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (−0.564 − 0.120i)3-s + (0.334 + 0.371i)4-s + (−0.0754 + 0.717i)5-s + (0.330 + 0.239i)6-s + (−0.706 − 0.707i)7-s + (−0.109 − 0.336i)8-s + (0.304 + 0.135i)9-s + (0.255 − 0.441i)10-s + (0.999 + 0.0145i)11-s + (−0.144 − 0.249i)12-s + (0.0595 − 0.0432i)13-s + (0.252 + 0.660i)14-s + (0.128 − 0.396i)15-s + (−0.0261 + 0.248i)16-s + (−1.20 + 0.536i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.256 - 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.471787 + 0.362724i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.471787 + 0.362724i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 + 0.406i)T \) |
| 3 | \( 1 + (0.978 + 0.207i)T \) |
| 7 | \( 1 + (1.86 + 1.87i)T \) |
| 11 | \( 1 + (-3.31 - 0.0483i)T \) |
| good | 5 | \( 1 + (0.168 - 1.60i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (-0.214 + 0.156i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (4.96 - 2.21i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (2.85 - 3.17i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (-4.45 - 7.71i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (3.02 - 9.30i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (0.442 + 4.21i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.958 - 0.203i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.45 - 7.56i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + (-5.38 + 5.97i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.835 - 7.95i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-8.38 - 9.31i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.758 - 7.21i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-3.08 + 5.34i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.52 + 4.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.88 + 6.53i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-10.1 - 4.49i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (7.60 + 5.52i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (7.87 + 13.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (14.3 - 10.4i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.99218578532372283737897623774, −10.55802618685982854608132749638, −9.541904440713702484659513306805, −8.728762743770633389506869803787, −7.27893826139973849166387800649, −6.86219142413443752422967530043, −5.92030349601172974437247229280, −4.19311535820921444166908230574, −3.22854822450850210417011981053, −1.47749204319939045409481506553,
0.50881052082589355232547555438, 2.39578298231796145271049034805, 4.23600722241166506193983841610, 5.24077680885648403073220865071, 6.47798979716960072906161996145, 6.84654940146639287439767504562, 8.559459239292622044979474043139, 8.950823479933301051165490950136, 9.753830146487740040697324636013, 10.89349498483553770029079887542
