L(s) = 1 | + (0.866 + 0.5i)2-s + (0.866 − 0.5i)3-s + (0.499 + 0.866i)4-s + (−2.23 − 0.133i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 − 0.866i)9-s + (−1.86 − 1.23i)10-s + (1 + 1.73i)11-s + (0.866 + 0.499i)12-s − 2i·13-s + (−1.99 + i)15-s + (−0.5 + 0.866i)16-s + (6.92 − 4i)17-s + (0.866 − 0.499i)18-s + (1 − 1.73i)19-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.499 − 0.288i)3-s + (0.249 + 0.433i)4-s + (−0.998 − 0.0599i)5-s + 0.408·6-s + 0.353i·8-s + (0.166 − 0.288i)9-s + (−0.590 − 0.389i)10-s + (0.301 + 0.522i)11-s + (0.249 + 0.144i)12-s − 0.554i·13-s + (−0.516 + 0.258i)15-s + (−0.125 + 0.216i)16-s + (1.68 − 0.970i)17-s + (0.204 − 0.117i)18-s + (0.229 − 0.397i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.603392174\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.603392174\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 + (-0.866 + 0.5i)T \) |
| 5 | \( 1 + (2.23 + 0.133i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 + (-6.92 + 4i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 + 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + (-3 - 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.92 - 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + (3.46 + 2i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.73 - i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5 + 8.66i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 + 6i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 14T + 71T^{2} \) |
| 73 | \( 1 + (-8.66 + 5i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16iT - 83T^{2} \) |
| 89 | \( 1 + (5 - 8.66i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10iT - 97T^{2} \) |
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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
−9.473409944480963892373182241593, −8.350727372728367329063559234535, −7.893800889365649907057890576913, −7.16075800944663059028857148940, −6.44752684069559325672638318265, −5.17354302103619018555649983099, −4.54651294350851055085063510324, −3.37154735280257540888558299612, −2.86603298045148701926221149402, −1.06934739624880270010609536850,
1.13970801905657607579293357361, 2.63640696935963794012093175947, 3.66663733573632959266708436034, 4.02536167082523644125823556723, 5.15559552920161069563638225767, 6.10295972489202503648616269027, 7.07639072215368640790287408328, 8.039948772536087059208080052550, 8.497581656739039398472329255897, 9.668597903753014524762801512805