L(s) = 1 | + (0.866 + 0.5i)2-s + (0.5 + 0.866i)3-s + (0.499 + 0.866i)4-s + (−2.62 − 1.51i)5-s + 0.999i·6-s + (−2.43 + 1.04i)7-s + 0.999i·8-s + (−0.499 + 0.866i)9-s + (−1.51 − 2.62i)10-s + (−3.75 + 2.16i)11-s + (−0.499 + 0.866i)12-s + (−0.613 + 3.55i)13-s + (−2.62 − 0.311i)14-s − 3.02i·15-s + (−0.5 + 0.866i)16-s + (−0.721 − 1.25i)17-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.288 + 0.499i)3-s + (0.249 + 0.433i)4-s + (−1.17 − 0.676i)5-s + 0.408i·6-s + (−0.918 + 0.394i)7-s + 0.353i·8-s + (−0.166 + 0.288i)9-s + (−0.478 − 0.829i)10-s + (−1.13 + 0.653i)11-s + (−0.144 + 0.249i)12-s + (−0.170 + 0.985i)13-s + (−0.702 − 0.0831i)14-s − 0.781i·15-s + (−0.125 + 0.216i)16-s + (−0.175 − 0.303i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.974 - 0.223i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.100748 + 0.889549i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.100748 + 0.889549i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (2.43 - 1.04i)T \) |
| 13 | \( 1 + (0.613 - 3.55i)T \) |
good | 5 | \( 1 + (2.62 + 1.51i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (3.75 - 2.16i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (0.721 + 1.25i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.12 - 0.652i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (2.36 - 4.09i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3.42T + 29T^{2} \) |
| 31 | \( 1 + (-1.46 + 0.847i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-9.89 - 5.71i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 8.53iT - 41T^{2} \) |
| 43 | \( 1 + 5.10T + 43T^{2} \) |
| 47 | \( 1 + (7.28 + 4.20i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.05 - 3.55i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.34 + 2.50i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.505 - 0.875i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.412 + 0.238i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 + (3.42 - 1.97i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.34 - 12.7i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 17.0iT - 83T^{2} \) |
| 89 | \( 1 + (-6.04 - 3.48i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 15.9iT - 97T^{2} \) |
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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
−11.60195421014295806213942801372, −10.20259287418231875606996100014, −9.408670035677477996880601611682, −8.414114724787995875698581805010, −7.67323532517915497455873635400, −6.72185420133081270642992363526, −5.40965232171126646076090511965, −4.55554683032370386124958395559, −3.73293120105644044749066875012, −2.55456120064573480752346567670,
0.38611898093966015570484685519, 2.79486675981860161952376302015, 3.24748908345429552309136182300, 4.43726327862997785776807878461, 5.86528192510412566388684837043, 6.73171930122222618811201736044, 7.71320517587277683734309953215, 8.270561491040753855771462293236, 9.811403794652546500363867206947, 10.60510016782524882822127191293