| L(s) = 1 | + (1.32 − 0.490i)2-s + (0.0490 − 0.998i)3-s + (1.51 − 1.30i)4-s + (0.251 − 0.420i)5-s + (−0.424 − 1.34i)6-s + (−0.848 + 1.03i)7-s + (1.37 − 2.46i)8-s + (−0.995 − 0.0980i)9-s + (0.128 − 0.681i)10-s + (2.04 + 0.731i)11-s + (−1.22 − 1.58i)12-s + (1.09 − 4.36i)13-s + (−0.618 + 1.78i)14-s + (−0.407 − 0.272i)15-s + (0.618 − 3.95i)16-s + (−0.860 + 0.574i)17-s + ⋯ |
| L(s) = 1 | + (0.938 − 0.346i)2-s + (0.0283 − 0.576i)3-s + (0.759 − 0.650i)4-s + (0.112 − 0.187i)5-s + (−0.173 − 0.550i)6-s + (−0.320 + 0.390i)7-s + (0.487 − 0.873i)8-s + (−0.331 − 0.0326i)9-s + (0.0405 − 0.215i)10-s + (0.616 + 0.220i)11-s + (−0.353 − 0.456i)12-s + (0.303 − 1.21i)13-s + (−0.165 + 0.477i)14-s + (−0.105 − 0.0703i)15-s + (0.154 − 0.987i)16-s + (−0.208 + 0.139i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.84791 - 2.02124i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.84791 - 2.02124i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.32 + 0.490i)T \) |
| 3 | \( 1 + (-0.0490 + 0.998i)T \) |
| good | 5 | \( 1 + (-0.251 + 0.420i)T + (-2.35 - 4.40i)T^{2} \) |
| 7 | \( 1 + (0.848 - 1.03i)T + (-1.36 - 6.86i)T^{2} \) |
| 11 | \( 1 + (-2.04 - 0.731i)T + (8.50 + 6.97i)T^{2} \) |
| 13 | \( 1 + (-1.09 + 4.36i)T + (-11.4 - 6.12i)T^{2} \) |
| 17 | \( 1 + (0.860 - 0.574i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (0.938 + 6.32i)T + (-18.1 + 5.51i)T^{2} \) |
| 23 | \( 1 + (0.210 + 0.693i)T + (-19.1 + 12.7i)T^{2} \) |
| 29 | \( 1 + (-1.09 + 2.31i)T + (-18.3 - 22.4i)T^{2} \) |
| 31 | \( 1 + (-2.91 - 7.04i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (4.33 - 3.21i)T + (10.7 - 35.4i)T^{2} \) |
| 41 | \( 1 + (-4.47 + 8.37i)T + (-22.7 - 34.0i)T^{2} \) |
| 43 | \( 1 + (10.2 - 0.503i)T + (42.7 - 4.21i)T^{2} \) |
| 47 | \( 1 + (1.74 - 8.76i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-0.564 - 1.19i)T + (-33.6 + 40.9i)T^{2} \) |
| 59 | \( 1 + (-0.794 - 3.16i)T + (-52.0 + 27.8i)T^{2} \) |
| 61 | \( 1 + (-2.79 + 2.53i)T + (5.97 - 60.7i)T^{2} \) |
| 67 | \( 1 + (-9.82 - 10.8i)T + (-6.56 + 66.6i)T^{2} \) |
| 71 | \( 1 + (-0.649 - 6.59i)T + (-69.6 + 13.8i)T^{2} \) |
| 73 | \( 1 + (-0.148 - 0.181i)T + (-14.2 + 71.5i)T^{2} \) |
| 79 | \( 1 + (6.19 - 1.23i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-13.8 - 10.2i)T + (24.0 + 79.4i)T^{2} \) |
| 89 | \( 1 + (-4.07 + 13.4i)T + (-74.0 - 49.4i)T^{2} \) |
| 97 | \( 1 + (-9.41 + 3.90i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.33150240465910089475484318980, −9.280771655614685382843327522779, −8.408818790982642024664983339824, −7.14644218290808707453945460251, −6.51832687686545926568484354968, −5.57496076104608567409619547625, −4.73560288026744597524071344026, −3.40063576831341274975851730470, −2.51949323540462922095648626805, −1.11119226346857555189416410243,
2.00975963569259489306518017372, 3.48394194024412482619478645262, 4.06512672952531502294246806960, 5.05710487964862109875528765058, 6.30050301850134317250794707607, 6.63606347056337423070520982352, 7.898140879303167905853304189117, 8.760248571262313221505005993235, 9.818583743027248878701785580253, 10.61409282670451674898917589180
