| L(s) = 1 | + 2-s − 3·3-s + 4-s − 2·5-s − 3·6-s + 8-s + 6·9-s − 2·10-s − 11-s − 3·12-s + 7·13-s + 6·15-s + 16-s − 2·17-s + 6·18-s − 2·20-s − 22-s − 8·23-s − 3·24-s − 25-s + 7·26-s − 9·27-s − 5·29-s + 6·30-s − 4·31-s + 32-s + 3·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.73·3-s + 1/2·4-s − 0.894·5-s − 1.22·6-s + 0.353·8-s + 2·9-s − 0.632·10-s − 0.301·11-s − 0.866·12-s + 1.94·13-s + 1.54·15-s + 1/4·16-s − 0.485·17-s + 1.41·18-s − 0.447·20-s − 0.213·22-s − 1.66·23-s − 0.612·24-s − 1/5·25-s + 1.37·26-s − 1.73·27-s − 0.928·29-s + 1.09·30-s − 0.718·31-s + 0.176·32-s + 0.522·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 7 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.822171020404918293241574164680, −8.406201341554071214442008999603, −7.59001708095982935582996619965, −6.54204507036176471515809041147, −6.02860827782459493657287753528, −5.24362529829616318653741665156, −4.19558443542356955904809517882, −3.64499829703371390584081659599, −1.61277139597212650066132551280, 0,
1.61277139597212650066132551280, 3.64499829703371390584081659599, 4.19558443542356955904809517882, 5.24362529829616318653741665156, 6.02860827782459493657287753528, 6.54204507036176471515809041147, 7.59001708095982935582996619965, 8.406201341554071214442008999603, 9.822171020404918293241574164680
