| L(s) = 1 | − 3·3-s + 5-s + 6·9-s + 6·13-s − 3·15-s + 4·17-s + 6·19-s − 3·23-s − 4·25-s − 9·27-s + 4·29-s + 9·31-s + 7·37-s − 18·39-s + 2·41-s + 6·43-s + 6·45-s − 12·47-s − 7·49-s − 12·51-s + 2·53-s − 18·57-s − 9·59-s − 8·61-s + 6·65-s + 15·67-s + 9·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.447·5-s + 2·9-s + 1.66·13-s − 0.774·15-s + 0.970·17-s + 1.37·19-s − 0.625·23-s − 4/5·25-s − 1.73·27-s + 0.742·29-s + 1.61·31-s + 1.15·37-s − 2.88·39-s + 0.312·41-s + 0.914·43-s + 0.894·45-s − 1.75·47-s − 49-s − 1.68·51-s + 0.274·53-s − 2.38·57-s − 1.17·59-s − 1.02·61-s + 0.744·65-s + 1.83·67-s + 1.08·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.380528696\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.380528696\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 15 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 5 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−8.287560885566275927735371889188, −7.72443767762056096735353713439, −6.63853162271313407088541674158, −6.11580963830131008904171742259, −5.70492148525469791258011898858, −4.90887653551788715606253711606, −4.08228664570882592197319840331, −3.07464307282927619124096681008, −1.48674422784208859966308566870, −0.838602038950360497584616436037,
0.838602038950360497584616436037, 1.48674422784208859966308566870, 3.07464307282927619124096681008, 4.08228664570882592197319840331, 4.90887653551788715606253711606, 5.70492148525469791258011898858, 6.11580963830131008904171742259, 6.63853162271313407088541674158, 7.72443767762056096735353713439, 8.287560885566275927735371889188
