| L(s) = 1 | + 3-s + 5-s − 4·7-s + 9-s − 4·11-s − 6·13-s + 15-s + 2·17-s − 4·19-s − 4·21-s + 25-s + 27-s − 10·29-s − 4·31-s − 4·33-s − 4·35-s + 10·37-s − 6·39-s + 2·41-s + 4·43-s + 45-s + 8·47-s + 9·49-s + 2·51-s − 2·53-s − 4·55-s − 4·57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 1.20·11-s − 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.872·21-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 0.718·31-s − 0.696·33-s − 0.676·35-s + 1.64·37-s − 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.149·45-s + 1.16·47-s + 9/7·49-s + 0.280·51-s − 0.274·53-s − 0.539·55-s − 0.529·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−9.622099009253011006773905670291, −9.053153555577834950733808605433, −7.73435767676089003755122228124, −7.26992741343389521999119886505, −6.14269520510095343624413187280, −5.33556450763905259522444606662, −4.10961287212778855632833672573, −2.91524621673910645918598375111, −2.28860875976801017534156160572, 0,
2.28860875976801017534156160572, 2.91524621673910645918598375111, 4.10961287212778855632833672573, 5.33556450763905259522444606662, 6.14269520510095343624413187280, 7.26992741343389521999119886505, 7.73435767676089003755122228124, 9.053153555577834950733808605433, 9.622099009253011006773905670291
