L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.725 − 0.418i)5-s − 0.999·6-s + (2.44 − 1.02i)7-s − 0.999i·8-s + (0.499 + 0.866i)9-s + (0.418 − 0.725i)10-s + (2.45 + 2.22i)11-s + (−0.866 + 0.499i)12-s − 2.59·13-s + (1.60 − 2.10i)14-s − 0.837·15-s + (−0.5 − 0.866i)16-s + (2.98 − 5.17i)17-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.324 − 0.187i)5-s − 0.408·6-s + (0.922 − 0.386i)7-s − 0.353i·8-s + (0.166 + 0.288i)9-s + (0.132 − 0.229i)10-s + (0.741 + 0.670i)11-s + (−0.249 + 0.144i)12-s − 0.719·13-s + (0.428 − 0.562i)14-s − 0.216·15-s + (−0.125 − 0.216i)16-s + (0.724 − 1.25i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56742 - 1.14189i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56742 - 1.14189i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.44 + 1.02i)T \) |
| 11 | \( 1 + (-2.45 - 2.22i)T \) |
good | 5 | \( 1 + (-0.725 + 0.418i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 2.59T + 13T^{2} \) |
| 17 | \( 1 + (-2.98 + 5.17i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.55 + 2.69i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.43 + 2.48i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.38iT - 29T^{2} \) |
| 31 | \( 1 + (0.913 + 0.527i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.49 - 9.51i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 1.27iT - 43T^{2} \) |
| 47 | \( 1 + (10.6 - 6.12i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.58 - 4.48i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (8.38 + 4.84i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.03 - 3.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.51 - 11.2i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14.0T + 71T^{2} \) |
| 73 | \( 1 + (4.95 - 8.58i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (11.7 - 6.75i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.99T + 83T^{2} \) |
| 89 | \( 1 + (-7.28 + 4.20i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 0.786iT - 97T^{2} \) |
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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
−11.28635072403398923742471688748, −10.01244396240604651950774396499, −9.418570586526654782379077978015, −7.88284206456802269797108887597, −7.10431778815556685930897780059, −6.03673253527668138070022940474, −4.91989696089540567693745559581, −4.35273086023382463965386997970, −2.55486941381222994980286187477, −1.23320182942830634192487966893,
1.85866897173974893527635867886, 3.55498297095056633872746056484, 4.57914314895432267115760475347, 5.71635855989185920171823146257, 6.16669765006712712036832407351, 7.51896918246085687039197636549, 8.374476977510867936466599022135, 9.461606857768660366405106443114, 10.56479082779166383218809773277, 11.28137773041310755513976620606