| L(s) = 1 | − 5-s − 3·7-s + 2·11-s − 2·13-s + 7·17-s − 6·19-s − 23-s + 25-s + 9·29-s + 9·31-s + 3·35-s − 7·37-s − 5·41-s − 8·47-s + 2·49-s + 11·53-s − 2·55-s − 9·59-s + 2·65-s − 3·67-s − 3·71-s − 6·73-s − 6·77-s − 8·79-s − 5·83-s − 7·85-s + 6·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 0.603·11-s − 0.554·13-s + 1.69·17-s − 1.37·19-s − 0.208·23-s + 1/5·25-s + 1.67·29-s + 1.61·31-s + 0.507·35-s − 1.15·37-s − 0.780·41-s − 1.16·47-s + 2/7·49-s + 1.51·53-s − 0.269·55-s − 1.17·59-s + 0.248·65-s − 0.366·67-s − 0.356·71-s − 0.702·73-s − 0.683·77-s − 0.900·79-s − 0.548·83-s − 0.759·85-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 11 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−8.198352029776081704179962583997, −7.20665802211394294788507076701, −6.58224189435894862107686196920, −6.03471583126998769636769855173, −4.98010725817959605569098291710, −4.18959516479436501828774024575, −3.33632083290262564079618868160, −2.70081086208938402175006033537, −1.28374529154109134819146796964, 0,
1.28374529154109134819146796964, 2.70081086208938402175006033537, 3.33632083290262564079618868160, 4.18959516479436501828774024575, 4.98010725817959605569098291710, 6.03471583126998769636769855173, 6.58224189435894862107686196920, 7.20665802211394294788507076701, 8.198352029776081704179962583997
