| L(s) = 1 | + (2.64 − 1.41i)3-s − 5.65i·5-s − 4·7-s + (5 − 7.48i)9-s + 8.48i·11-s − 10.5·13-s + (−8.00 − 14.9i)15-s − 14.9i·17-s − 5.29·19-s + (−10.5 + 5.65i)21-s − 29.9i·23-s − 7.00·25-s + (2.64 − 26.8i)27-s − 16.9i·29-s − 4·31-s + ⋯ |
| L(s) = 1 | + (0.881 − 0.471i)3-s − 1.13i·5-s − 0.571·7-s + (0.555 − 0.831i)9-s + 0.771i·11-s − 0.814·13-s + (−0.533 − 0.997i)15-s − 0.880i·17-s − 0.278·19-s + (−0.503 + 0.269i)21-s − 1.30i·23-s − 0.280·25-s + (0.0979 − 0.995i)27-s − 0.585i·29-s − 0.129·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.881 + 0.471i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.605726555\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.605726555\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.64 + 1.41i)T \) |
| good | 5 | \( 1 + 5.65iT - 25T^{2} \) |
| 7 | \( 1 + 4T + 49T^{2} \) |
| 11 | \( 1 - 8.48iT - 121T^{2} \) |
| 13 | \( 1 + 10.5T + 169T^{2} \) |
| 17 | \( 1 + 14.9iT - 289T^{2} \) |
| 19 | \( 1 + 5.29T + 361T^{2} \) |
| 23 | \( 1 + 29.9iT - 529T^{2} \) |
| 29 | \( 1 + 16.9iT - 841T^{2} \) |
| 31 | \( 1 + 4T + 961T^{2} \) |
| 37 | \( 1 + 52.9T + 1.36e3T^{2} \) |
| 41 | \( 1 - 29.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 5.29T + 1.84e3T^{2} \) |
| 47 | \( 1 - 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 48.0iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 95.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 47.6T + 4.48e3T^{2} \) |
| 71 | \( 1 - 89.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 6T + 5.32e3T^{2} \) |
| 79 | \( 1 - 124T + 6.24e3T^{2} \) |
| 83 | \( 1 - 2.82iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 104. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 118T + 9.40e3T^{2} \) |
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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
−9.573734039105419132870252499358, −8.924428371748251931538146609969, −8.146478767079624363347779491205, −7.22682624701312104027331158304, −6.47759112152037158505955840955, −5.04756044315865994921284032518, −4.30629836065034934271932819234, −2.96637157196957986984160080648, −1.90584818600084576729480232401, −0.46013982873005211387768946216,
1.99331429114939353731127718800, 3.18963359612246228769589600916, 3.61130164447761826766408356088, 5.03990659343914896806053830633, 6.21588934381918846409323705971, 7.12439590654964477904521391105, 7.88248887739382857087655179914, 8.897089070068994630246049757798, 9.626278480743209916150550510964, 10.53117639060414212858644677116
