| L(s) = 1 | + 3-s + 9-s − 4·11-s + 2·13-s − 6·17-s + 4·19-s − 8·23-s + 27-s − 6·29-s − 4·33-s − 10·37-s + 2·39-s − 6·41-s − 43-s − 8·47-s − 7·49-s − 6·51-s − 2·53-s + 4·57-s − 4·59-s + 14·61-s + 4·67-s − 8·69-s − 16·71-s − 10·73-s + 8·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 1.20·11-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 1.66·23-s + 0.192·27-s − 1.11·29-s − 0.696·33-s − 1.64·37-s + 0.320·39-s − 0.937·41-s − 0.152·43-s − 1.16·47-s − 49-s − 0.840·51-s − 0.274·53-s + 0.529·57-s − 0.520·59-s + 1.79·61-s + 0.488·67-s − 0.963·69-s − 1.89·71-s − 1.17·73-s + 0.900·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 43 | \( 1 + T \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−13.52084115349458, −13.11516774979905, −12.77303585012310, −12.04567640136834, −11.63645260097013, −11.15879167697824, −10.66185360249121, −10.11905271706910, −9.836338856022358, −9.235014542254089, −8.668483179800299, −8.308670776779834, −7.909266472261080, −7.343200170043675, −6.856720598760746, −6.362738762704853, −5.670919308645741, −5.278516260026130, −4.694492292954042, −4.105574238831314, −3.536942410914320, −3.098433231701626, −2.399544626190368, −1.862315301987673, −1.414481156402592, 0, 0,
1.414481156402592, 1.862315301987673, 2.399544626190368, 3.098433231701626, 3.536942410914320, 4.105574238831314, 4.694492292954042, 5.278516260026130, 5.670919308645741, 6.362738762704853, 6.856720598760746, 7.343200170043675, 7.909266472261080, 8.308670776779834, 8.668483179800299, 9.235014542254089, 9.836338856022358, 10.11905271706910, 10.66185360249121, 11.15879167697824, 11.63645260097013, 12.04567640136834, 12.77303585012310, 13.11516774979905, 13.52084115349458
