L(s) = 1 | + (0.142 − 1.63i)5-s + (−0.939 + 0.342i)9-s + (1.75 − 0.816i)13-s + (0.0736 − 0.157i)17-s + (−1.65 − 0.292i)25-s + (−0.816 + 0.218i)29-s + (0.342 − 0.939i)37-s + (0.524 − 1.43i)41-s + (0.424 + 1.58i)45-s + (−0.173 + 0.984i)49-s + (−1.32 − 1.11i)53-s + (−0.766 − 1.64i)61-s + (−1.08 − 2.97i)65-s + i·73-s + (0.766 − 0.642i)81-s + ⋯ |
L(s) = 1 | + (0.142 − 1.63i)5-s + (−0.939 + 0.342i)9-s + (1.75 − 0.816i)13-s + (0.0736 − 0.157i)17-s + (−1.65 − 0.292i)25-s + (−0.816 + 0.218i)29-s + (0.342 − 0.939i)37-s + (0.524 − 1.43i)41-s + (0.424 + 1.58i)45-s + (−0.173 + 0.984i)49-s + (−1.32 − 1.11i)53-s + (−0.766 − 1.64i)61-s + (−1.08 − 2.97i)65-s + i·73-s + (0.766 − 0.642i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0237 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.132672607\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.132672607\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 37 | \( 1 + (-0.342 + 0.939i)T \) |
good | 3 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 5 | \( 1 + (-0.142 + 1.63i)T + (-0.984 - 0.173i)T^{2} \) |
| 7 | \( 1 + (0.173 - 0.984i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 0.816i)T + (0.642 - 0.766i)T^{2} \) |
| 17 | \( 1 + (-0.0736 + 0.157i)T + (-0.642 - 0.766i)T^{2} \) |
| 19 | \( 1 + (0.342 + 0.939i)T^{2} \) |
| 23 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 29 | \( 1 + (0.816 - 0.218i)T + (0.866 - 0.5i)T^{2} \) |
| 31 | \( 1 + iT^{2} \) |
| 41 | \( 1 + (-0.524 + 1.43i)T + (-0.766 - 0.642i)T^{2} \) |
| 43 | \( 1 - iT^{2} \) |
| 47 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (1.32 + 1.11i)T + (0.173 + 0.984i)T^{2} \) |
| 59 | \( 1 + (0.984 - 0.173i)T^{2} \) |
| 61 | \( 1 + (0.766 + 1.64i)T + (-0.642 + 0.766i)T^{2} \) |
| 67 | \( 1 + (-0.173 + 0.984i)T^{2} \) |
| 71 | \( 1 + (-0.939 + 0.342i)T^{2} \) |
| 73 | \( 1 - iT - T^{2} \) |
| 79 | \( 1 + (-0.984 - 0.173i)T^{2} \) |
| 83 | \( 1 + (0.766 - 0.642i)T^{2} \) |
| 89 | \( 1 + (-0.173 - 1.98i)T + (-0.984 + 0.173i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 0.469i)T + (0.866 + 0.5i)T^{2} \) |
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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
−9.104343716696493956562342740767, −8.135326520358907180846239646064, −7.902324106316434319359416443864, −6.41998261824713632365039117558, −5.59102191366050918339771787016, −5.25983954880819794668096294515, −4.15327978145002244599892271337, −3.30721540438529872019440115631, −1.93760895102200272764976407752, −0.796957254173391798857415290309,
1.67382314886382594167835765077, 2.91325130672881974487510345831, 3.41638985942806182652361679867, 4.39590503361247192684467944615, 5.96820552667795383434246602447, 6.10444395230467232854845777787, 6.90575272034489007779893859858, 7.79436087751693092087540514705, 8.608604289092814368869222453222, 9.337594484474295683235223987968