L(s) = 1 | + (0.866 − 0.5i)2-s + (0.499 − 0.866i)4-s + 2.44·5-s + (2.62 + 0.358i)7-s − 0.999i·8-s + (2.12 − 1.22i)10-s − 4.24i·11-s + (0.621 − 0.358i)13-s + (2.44 − i)14-s + (−0.5 − 0.866i)16-s + (−1.22 − 2.12i)17-s + (−4.24 − 2.44i)19-s + (1.22 − 2.12i)20-s + (−2.12 − 3.67i)22-s + 6i·23-s + ⋯ |
L(s) = 1 | + (0.612 − 0.353i)2-s + (0.249 − 0.433i)4-s + 1.09·5-s + (0.990 + 0.135i)7-s − 0.353i·8-s + (0.670 − 0.387i)10-s − 1.27i·11-s + (0.172 − 0.0994i)13-s + (0.654 − 0.267i)14-s + (−0.125 − 0.216i)16-s + (−0.297 − 0.514i)17-s + (−0.973 − 0.561i)19-s + (0.273 − 0.474i)20-s + (−0.452 − 0.783i)22-s + 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.572 + 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.048931910\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.048931910\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.358i)T \) |
good | 5 | \( 1 - 2.44T + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.358i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.22 + 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 + 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6iT - 23T^{2} \) |
| 29 | \( 1 + (-1.52 - 0.878i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.22 - 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (6.42 + 11.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-12.5 + 7.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.22 + 2.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.62 - 2.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.74 - 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (4.75 - 2.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 + 0.655i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (7.64 - 13.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (1.52 - 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.74 - 1.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.907329727569454197009059744217, −8.808155411483527225324782436851, −8.290024689611512236877090797614, −6.93666157987769694645246082764, −6.10616477558921482681492317146, −5.36033508876271559204142493402, −4.64578173476671596624594088686, −3.33363051843523330026535384511, −2.31635502052923610000212579978, −1.24410908931344260854396533071,
1.75284767361655819494280799994, 2.44302344720756055155918673236, 4.22003702026723668411839997607, 4.64861845571012709943940239957, 5.78686310651421565899112708017, 6.40301772174630998586028001174, 7.34983428272012878206815206113, 8.245510953505841040792592914807, 9.001023353432801534927346230556, 10.17336668627393862377756689113