| L(s) = 1 | − 2·3-s + 9-s − 11-s + 4·13-s + 4·23-s + 4·27-s − 29-s + 2·31-s + 2·33-s − 10·37-s − 8·39-s + 6·41-s − 2·43-s − 5·47-s − 7·49-s + 53-s − 3·59-s − 9·61-s + 4·67-s − 8·69-s + 12·71-s − 73-s − 11·79-s − 11·81-s − 14·83-s + 2·87-s + 5·89-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/3·9-s − 0.301·11-s + 1.10·13-s + 0.834·23-s + 0.769·27-s − 0.185·29-s + 0.359·31-s + 0.348·33-s − 1.64·37-s − 1.28·39-s + 0.937·41-s − 0.304·43-s − 0.729·47-s − 49-s + 0.137·53-s − 0.390·59-s − 1.15·61-s + 0.488·67-s − 0.963·69-s + 1.42·71-s − 0.117·73-s − 1.23·79-s − 1.22·81-s − 1.53·83-s + 0.214·87-s + 0.529·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 317900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 9 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 5 T + p T^{2} \) |
| 97 | \( 1 - 8 T + p T^{2} \) |
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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
−12.77414255341214, −12.43052947088324, −11.82112447343085, −11.45861424283207, −11.05520437855773, −10.72819797080464, −10.29165488576285, −9.789995676102684, −9.133245175175741, −8.792441059823066, −8.193957364510511, −7.875264034639427, −6.988381854373179, −6.844970421399752, −6.235134803268806, −5.798219710274808, −5.413886240720437, −4.850608495822684, −4.492233796760500, −3.746638768954093, −3.220741376409467, −2.756327060647143, −1.838109733344744, −1.339250414274537, −0.6738776062633068, 0,
0.6738776062633068, 1.339250414274537, 1.838109733344744, 2.756327060647143, 3.220741376409467, 3.746638768954093, 4.492233796760500, 4.850608495822684, 5.413886240720437, 5.798219710274808, 6.235134803268806, 6.844970421399752, 6.988381854373179, 7.875264034639427, 8.193957364510511, 8.792441059823066, 9.133245175175741, 9.789995676102684, 10.29165488576285, 10.72819797080464, 11.05520437855773, 11.45861424283207, 11.82112447343085, 12.43052947088324, 12.77414255341214
