L(s) = 1 | − 3-s − 7-s + 9-s + 4·11-s + 2·13-s − 2·17-s + 4·19-s + 21-s − 27-s − 10·29-s − 4·33-s − 6·37-s − 2·39-s − 6·41-s − 4·43-s − 8·47-s + 49-s + 2·51-s − 6·53-s − 4·57-s + 4·59-s − 10·61-s − 63-s + 4·67-s + 16·71-s + 14·73-s − 4·77-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.485·17-s + 0.917·19-s + 0.218·21-s − 0.192·27-s − 1.85·29-s − 0.696·33-s − 0.986·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 1.16·47-s + 1/7·49-s + 0.280·51-s − 0.824·53-s − 0.529·57-s + 0.520·59-s − 1.28·61-s − 0.125·63-s + 0.488·67-s + 1.89·71-s + 1.63·73-s − 0.455·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.24601859955624368959603630077, −6.70277321216489461232070791272, −6.16513802484153015112031956328, −5.39725647850774733963544801765, −4.73863842780506411063344379916, −3.68979269296572291917707335052, −3.43131613455748210570172680022, −1.99674106072445845327308982099, −1.24290743460620606763533635146, 0,
1.24290743460620606763533635146, 1.99674106072445845327308982099, 3.43131613455748210570172680022, 3.68979269296572291917707335052, 4.73863842780506411063344379916, 5.39725647850774733963544801765, 6.16513802484153015112031956328, 6.70277321216489461232070791272, 7.24601859955624368959603630077