L(s) = 1 | + (−0.483 + 1.32i)3-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−1.39 − 0.245i)15-s + (0.766 − 0.642i)17-s + (−0.909 − 1.08i)21-s + (−0.909 + 1.08i)29-s + (−1.39 + 0.245i)33-s + (−0.939 − 0.342i)35-s − 1.41i·37-s + (0.483 − 1.32i)41-s + (−0.173 − 0.984i)43-s + (0.500 − 0.866i)45-s + ⋯ |
L(s) = 1 | + (−0.483 + 1.32i)3-s + (0.173 + 0.984i)5-s + (−0.5 + 0.866i)7-s + (−0.766 − 0.642i)9-s + (0.5 + 0.866i)11-s + (−1.39 − 0.245i)15-s + (0.766 − 0.642i)17-s + (−0.909 − 1.08i)21-s + (−0.909 + 1.08i)29-s + (−1.39 + 0.245i)33-s + (−0.939 − 0.342i)35-s − 1.41i·37-s + (0.483 − 1.32i)41-s + (−0.173 − 0.984i)43-s + (0.500 − 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.361i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8811116323\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8811116323\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (0.483 - 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 5 | \( 1 + (-0.173 - 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 17 | \( 1 + (-0.766 + 0.642i)T + (0.173 - 0.984i)T^{2} \) |
| 23 | \( 1 + (-0.939 - 0.342i)T^{2} \) |
| 29 | \( 1 + (0.909 - 1.08i)T + (-0.173 - 0.984i)T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + 1.41iT - T^{2} \) |
| 41 | \( 1 + (-0.483 + 1.32i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \) |
| 47 | \( 1 + (0.766 + 0.642i)T + (0.173 + 0.984i)T^{2} \) |
| 53 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 59 | \( 1 + (-0.909 - 1.08i)T + (-0.173 + 0.984i)T^{2} \) |
| 61 | \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \) |
| 67 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 71 | \( 1 + (-1.39 + 0.245i)T + (0.939 - 0.342i)T^{2} \) |
| 73 | \( 1 + (0.939 + 0.342i)T + (0.766 + 0.642i)T^{2} \) |
| 79 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 83 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 97 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
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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
−10.20738088278973735364599357793, −9.297630073011065757528156329055, −8.996425204523304239093638889217, −7.43561753448272157408539335310, −6.80659853360744515037459970363, −5.71498475348690877934152104775, −5.24417629095198029266280124561, −4.06780545935320820712956057492, −3.30577327770142786863982875005, −2.23528097981416535904261188038,
0.817372157785485705685165283926, 1.59500835757038432286176786786, 3.22985730346293703550190729864, 4.30271765384478820482109408846, 5.45172668806129151375356982729, 6.24955154244713574311108956183, 6.76068730644118514550441698590, 7.88828603675667764111646175745, 8.224774110297398425944578974103, 9.406453220467616224337690223636