L(s) = 1 | − 2-s + 4-s + (−2.05 + 3.56i)5-s + (−2.61 − 0.405i)7-s − 8-s + (2.05 − 3.56i)10-s + (2.02 − 3.50i)11-s + (1.81 − 3.11i)13-s + (2.61 + 0.405i)14-s + 16-s − 0.715·17-s + (1.92 + 3.33i)19-s + (−2.05 + 3.56i)20-s + (−2.02 + 3.50i)22-s + 4.09·23-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + (−0.920 + 1.59i)5-s + (−0.988 − 0.153i)7-s − 0.353·8-s + (0.650 − 1.12i)10-s + (0.609 − 1.05i)11-s + (0.503 − 0.864i)13-s + (0.698 + 0.108i)14-s + 0.250·16-s − 0.173·17-s + (0.441 + 0.764i)19-s + (−0.460 + 0.796i)20-s + (−0.431 + 0.746i)22-s + 0.854·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8364500409\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8364500409\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.61 + 0.405i)T \) |
| 13 | \( 1 + (-1.81 + 3.11i)T \) |
good | 5 | \( 1 + (2.05 - 3.56i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.02 + 3.50i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + 0.715T + 17T^{2} \) |
| 19 | \( 1 + (-1.92 - 3.33i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 4.09T + 23T^{2} \) |
| 29 | \( 1 + (4.50 + 7.80i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.82 - 3.16i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 + (-2.88 - 4.99i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.28 + 2.22i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.28 - 2.23i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.35 - 2.35i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 12.9T + 59T^{2} \) |
| 61 | \( 1 + (-4.71 - 8.16i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.583 + 1.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.10 - 8.83i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (1.25 + 2.17i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.70 - 11.6i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−9.590165899865978344780484951266, −8.601734453643114452131329599797, −7.77321402926671015152198419916, −7.24751511141492683362744008515, −6.23288052611920574100140516036, −6.00606257449477112189619606872, −4.01013051909118290983730402465, −3.29291029319103548485284626704, −2.74035423798191915412454769017, −0.75242563158121427054608501350,
0.66500642843144444556355764730, 1.80046979761412594804723960718, 3.35920907239434093819727246486, 4.29304817552260928301353340055, 5.04169097949422544825830478996, 6.24279620617545845286497558806, 7.11451440311504435464029649468, 7.71703489996302662772503004216, 8.912366566954177016580297674485, 9.100130871211293517989158264524