| L(s) = 1 | − 3-s − 2·5-s + 2·7-s + 9-s − 4·13-s + 2·15-s − 2·17-s + 4·19-s − 2·21-s − 4·23-s − 25-s − 27-s − 6·29-s + 2·31-s − 4·35-s − 8·37-s + 4·39-s − 2·41-s + 4·43-s − 2·45-s − 12·47-s − 3·49-s + 2·51-s − 6·53-s − 4·57-s − 4·59-s + 2·63-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s − 1.10·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.676·35-s − 1.31·37-s + 0.640·39-s − 0.312·41-s + 0.609·43-s − 0.298·45-s − 1.75·47-s − 3/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s − 0.520·59-s + 0.251·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−9.997326270238289527626106251734, −9.076892784593240226830796843651, −7.85430133808932302275185030514, −7.53733984796689177014830049764, −6.40576520020805405834867002447, −5.21555812892794413521744860012, −4.56793967647219176251846575918, −3.44980588384105099817944459966, −1.84821243380364994890921297905, 0,
1.84821243380364994890921297905, 3.44980588384105099817944459966, 4.56793967647219176251846575918, 5.21555812892794413521744860012, 6.40576520020805405834867002447, 7.53733984796689177014830049764, 7.85430133808932302275185030514, 9.076892784593240226830796843651, 9.997326270238289527626106251734
