L(s) = 1 | + (0.866 − 0.5i)2-s + (−0.5 + 0.866i)3-s + (0.499 − 0.866i)4-s + (2.59 − 1.5i)5-s + 0.999i·6-s − 0.999i·8-s + (1 + 1.73i)9-s + (1.5 − 2.59i)10-s + (0.499 + 0.866i)12-s + (−2 − 3i)13-s + 3i·15-s + (−0.5 − 0.866i)16-s + (1.5 − 2.59i)17-s + (1.73 + i)18-s + (5.19 − 3i)19-s − 3i·20-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (−0.288 + 0.499i)3-s + (0.249 − 0.433i)4-s + (1.16 − 0.670i)5-s + 0.408i·6-s − 0.353i·8-s + (0.333 + 0.577i)9-s + (0.474 − 0.821i)10-s + (0.144 + 0.249i)12-s + (−0.554 − 0.832i)13-s + 0.774i·15-s + (−0.125 − 0.216i)16-s + (0.363 − 0.630i)17-s + (0.408 + 0.235i)18-s + (1.19 − 0.688i)19-s − 0.670i·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1274 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.756 + 0.653i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.744505768\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.744505768\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 \) |
| 13 | \( 1 + (2 + 3i)T \) |
good | 3 | \( 1 + (0.5 - 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.59 + 1.5i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-5.19 + 3i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.59 + 1.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + (2.59 - 1.5i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 + 5.19i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.19 + 3i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 + 6.92i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.3 - 6i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15iT - 71T^{2} \) |
| 73 | \( 1 + (-5.19 - 3i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + (-5.19 + 3i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−9.687022319853740857327447430957, −9.237195650937738707766655380089, −7.86323168614981426475464671924, −7.04525810294726909293782920064, −5.75794983489438271745860993838, −5.20891264970158034807408831411, −4.80828917806803805810119054582, −3.41108451277968069361787984989, −2.34924094196406718654719032890, −1.12780256515364888903745408009,
1.46931876729662613608675789985, 2.56373061474021031544810769533, 3.65743169481239629494540953851, 4.82880216076096719655633768152, 5.83764207149067862193211377068, 6.38739812144408266708957296333, 7.01534732486635067279065776935, 7.81755528338631880810873813177, 9.072754156700747130232002672901, 9.795017444203280470936003003825