| L(s) = 1 | + (−0.913 − 0.406i)2-s + (−0.978 − 0.207i)3-s + (0.669 + 0.743i)4-s + (−0.435 + 4.14i)5-s + (0.809 + 0.587i)6-s + (2.64 − 0.0306i)7-s + (−0.309 − 0.951i)8-s + (0.913 + 0.406i)9-s + (2.08 − 3.60i)10-s + (−3.29 + 0.384i)11-s + (−0.499 − 0.866i)12-s + (−1.76 + 1.28i)13-s + (−2.42 − 1.04i)14-s + (1.28 − 3.95i)15-s + (−0.104 + 0.994i)16-s + (−0.794 + 0.353i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 − 0.287i)2-s + (−0.564 − 0.120i)3-s + (0.334 + 0.371i)4-s + (−0.194 + 1.85i)5-s + (0.330 + 0.239i)6-s + (0.999 − 0.0115i)7-s + (−0.109 − 0.336i)8-s + (0.304 + 0.135i)9-s + (0.658 − 1.14i)10-s + (−0.993 + 0.115i)11-s + (−0.144 − 0.249i)12-s + (−0.489 + 0.355i)13-s + (−0.649 − 0.280i)14-s + (0.332 − 1.02i)15-s + (−0.0261 + 0.248i)16-s + (−0.192 + 0.0858i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.620 - 0.784i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.246715 + 0.509460i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.246715 + 0.509460i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.64 + 0.0306i)T \) |
| 11 | \( 1 + (3.29 - 0.384i)T \) |
| good | 5 | \( 1 + (0.435 - 4.14i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (1.76 - 1.28i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.794 - 0.353i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.776 + 0.862i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.42 + 2.47i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.53 - 4.71i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-0.186 - 1.77i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (9.61 - 2.04i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (-2.10 - 6.48i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.87T + 43T^{2} \) |
| 47 | \( 1 + (5.05 - 5.61i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (0.574 + 5.46i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-5.44 - 6.05i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.0766 - 0.728i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (3.89 - 6.73i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.47 + 3.25i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.15 - 1.28i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (14.5 + 6.47i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.738 + 0.536i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (6.98 + 12.1i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-12.0 + 8.77i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.23146596811249186482936788423, −10.52188599108102503812421140083, −10.04631631600061821446917965241, −8.566978700451298340029745156989, −7.47315739359115934908635249793, −7.11179311379666616943999642166, −5.95315036353802658679595621441, −4.61595606667251853011357246815, −3.09108886376083451367680432961, −2.02827411723724501309987913336,
0.44901217289101126098697567957, 1.87126695541154793801563940493, 4.24864346592883391910256372476, 5.25040597384754709358526463182, 5.61756098678271441898760553350, 7.42883923889532476806617737763, 8.059731747040516924219326754840, 8.822003062929850467998324211863, 9.727794027300129810749812166038, 10.67130353212839300855043329333
