L(s) = 1 | + (0.5 − 0.866i)3-s − i·7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 1.5i)17-s + (−0.866 − 1.5i)19-s + (−0.866 − 0.5i)21-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 1.73·29-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s − 1.73·43-s + (−0.5 − 0.866i)47-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)3-s − i·7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + (0.866 − 1.5i)17-s + (−0.866 − 1.5i)19-s + (−0.866 − 0.5i)21-s + (0.5 + 0.866i)23-s + (−0.5 + 0.866i)25-s − 0.999·27-s + 1.73·29-s + (0.499 + 0.866i)33-s + (−0.5 − 0.866i)37-s − 1.73·43-s + (−0.5 − 0.866i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.553 + 0.832i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.300690582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.300690582\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + iT \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
good | 5 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - 1.73T + T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 1.73T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T + T^{2} \) |
| 73 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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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
−8.263588305079994415976722656258, −7.62324586365053348350668194762, −6.92237760627241118935667185512, −6.74952027003886635987124694789, −5.30526924293699133434012409122, −4.75814704802173571329156375224, −3.58607698464127728205398289690, −2.85288805362582951474512305504, −1.87038119668336463679930121489, −0.68715686154561397740834493025,
1.75002477874755516307708948911, 2.77768371298720228872312493242, 3.43538748100146618661554237542, 4.34097057678247319323641695110, 5.17905538987832589625644209466, 5.97000029188545252031442503920, 6.43053108477506393518111422516, 8.041399836620591840986619587124, 8.360505739599596312663017057904, 8.607638404773675300498756173038