| L(s) = 1 | + 3-s − 2·5-s − 4·7-s + 9-s − 11-s − 2·13-s − 2·15-s − 2·17-s − 4·21-s − 8·23-s − 25-s + 27-s − 6·29-s + 8·31-s − 33-s + 8·35-s + 6·37-s − 2·39-s − 2·41-s − 2·45-s − 8·47-s + 9·49-s − 2·51-s + 6·53-s + 2·55-s + 4·59-s + 6·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s − 1.51·7-s + 1/3·9-s − 0.301·11-s − 0.554·13-s − 0.516·15-s − 0.485·17-s − 0.872·21-s − 1.66·23-s − 1/5·25-s + 0.192·27-s − 1.11·29-s + 1.43·31-s − 0.174·33-s + 1.35·35-s + 0.986·37-s − 0.320·39-s − 0.312·41-s − 0.298·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s + 0.824·53-s + 0.269·55-s + 0.520·59-s + 0.768·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.05246880525653227640097293961, −9.723139098205724899141727122835, −8.548353092044889979587894449296, −7.75720782023225106841673661541, −6.87928263143909248442940323738, −5.90723738552130668805725208182, −4.35461089374668409702872860477, −3.53074178558790394542885743604, −2.45250023112523203194083612695, 0,
2.45250023112523203194083612695, 3.53074178558790394542885743604, 4.35461089374668409702872860477, 5.90723738552130668805725208182, 6.87928263143909248442940323738, 7.75720782023225106841673661541, 8.548353092044889979587894449296, 9.723139098205724899141727122835, 10.05246880525653227640097293961
