| L(s) = 1 | − 3-s − 2·5-s + 9-s − 11-s − 13-s + 2·15-s − 2·17-s + 4·19-s − 25-s − 27-s + 6·29-s − 8·31-s + 33-s + 2·37-s + 39-s + 6·41-s + 12·43-s − 2·45-s + 12·47-s − 7·49-s + 2·51-s − 10·53-s + 2·55-s − 4·57-s − 4·59-s − 6·61-s + 2·65-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1/3·9-s − 0.301·11-s − 0.277·13-s + 0.516·15-s − 0.485·17-s + 0.917·19-s − 1/5·25-s − 0.192·27-s + 1.11·29-s − 1.43·31-s + 0.174·33-s + 0.328·37-s + 0.160·39-s + 0.937·41-s + 1.82·43-s − 0.298·45-s + 1.75·47-s − 49-s + 0.280·51-s − 1.37·53-s + 0.269·55-s − 0.529·57-s − 0.520·59-s − 0.768·61-s + 0.248·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.56662994184935377233455908786, −7.07317858865885137256697453422, −6.13639229184493048945057738658, −5.52981570191461136896469464386, −4.67174955580657812703625980456, −4.12573688319162731524891089437, −3.24504107956546556770475011333, −2.32125217980865252265981286524, −1.05454401773367711170639235342, 0,
1.05454401773367711170639235342, 2.32125217980865252265981286524, 3.24504107956546556770475011333, 4.12573688319162731524891089437, 4.67174955580657812703625980456, 5.52981570191461136896469464386, 6.13639229184493048945057738658, 7.07317858865885137256697453422, 7.56662994184935377233455908786
