L(s) = 1 | + (1.02 + 0.592i)5-s − 4.00i·7-s + 3.72·11-s + (−0.163 − 0.284i)13-s + (−2.92 − 1.69i)17-s + (3.68 − 2.32i)19-s + (1.20 + 2.08i)23-s + (−1.79 − 3.11i)25-s + (0.884 − 0.510i)29-s − 0.252i·31-s + (2.37 − 4.11i)35-s − 2.26·37-s + (−6.04 − 3.48i)41-s + (0.904 + 0.522i)43-s + (3.48 + 6.03i)47-s + ⋯ |
L(s) = 1 | + (0.459 + 0.265i)5-s − 1.51i·7-s + 1.12·11-s + (−0.0454 − 0.0787i)13-s + (−0.710 − 0.410i)17-s + (0.846 − 0.533i)19-s + (0.251 + 0.435i)23-s + (−0.359 − 0.622i)25-s + (0.164 − 0.0948i)29-s − 0.0452i·31-s + (0.401 − 0.695i)35-s − 0.372·37-s + (−0.943 − 0.544i)41-s + (0.137 + 0.0796i)43-s + (0.508 + 0.880i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2736 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.158 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.930266860\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.930266860\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-3.68 + 2.32i)T \) |
good | 5 | \( 1 + (-1.02 - 0.592i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + 4.00iT - 7T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 + (0.163 + 0.284i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (2.92 + 1.69i)T + (8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-1.20 - 2.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-0.884 + 0.510i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 0.252iT - 31T^{2} \) |
| 37 | \( 1 + 2.26T + 37T^{2} \) |
| 41 | \( 1 + (6.04 + 3.48i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.904 - 0.522i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3.48 - 6.03i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-10.8 + 6.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.581 + 1.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (4.11 + 7.12i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (8.16 - 4.71i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.12 + 1.94i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.05 - 1.82i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.56 + 3.21i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 5.14T + 83T^{2} \) |
| 89 | \( 1 + (-3.86 + 2.23i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.12 - 7.14i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−8.771951641977785489475108425349, −7.72842291424386165098260714013, −6.99737531960699200159531428112, −6.63239682748133946404405896659, −5.59719889980727011769545863988, −4.57103328557198214277449982997, −3.92183446468181623665028734469, −3.00370184536956550378909081423, −1.72555185006904243991172557943, −0.65011914377798465208680188448,
1.39014749211946045700046493448, 2.23986760184049300149976204324, 3.27268653545623004494356779187, 4.29646722221783352698585187590, 5.30388523928787217816625708430, 5.85444973074110519591800282302, 6.56771265681633166834899924365, 7.44678222888247432958379999872, 8.678585658745172565423187750232, 8.810124790539458554177776996123