L(s) = 1 | + 5-s + 3·7-s − 5·11-s − 4·13-s − 8·17-s − 2·19-s + 2·23-s − 4·25-s − 6·29-s − 7·31-s + 3·35-s + 6·37-s − 6·41-s + 2·43-s + 6·47-s + 2·49-s − 5·53-s − 5·55-s + 4·59-s + 8·61-s − 4·65-s + 10·67-s − 8·71-s + 73-s − 15·77-s + 16·79-s + 11·83-s + ⋯ |
L(s) = 1 | + 0.447·5-s + 1.13·7-s − 1.50·11-s − 1.10·13-s − 1.94·17-s − 0.458·19-s + 0.417·23-s − 4/5·25-s − 1.11·29-s − 1.25·31-s + 0.507·35-s + 0.986·37-s − 0.937·41-s + 0.304·43-s + 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.674·55-s + 0.520·59-s + 1.02·61-s − 0.496·65-s + 1.22·67-s − 0.949·71-s + 0.117·73-s − 1.70·77-s + 1.80·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−8.956767999431597194519714231575, −8.022029533648621851025185775650, −7.50411880188001570794485201686, −6.57131885882624009354198232118, −5.39530272431252776298615511367, −4.98950710082664862043225418464, −4.03480598602149279339539607760, −2.43021211040984556747660649674, −2.02037094114478022385464815961, 0,
2.02037094114478022385464815961, 2.43021211040984556747660649674, 4.03480598602149279339539607760, 4.98950710082664862043225418464, 5.39530272431252776298615511367, 6.57131885882624009354198232118, 7.50411880188001570794485201686, 8.022029533648621851025185775650, 8.956767999431597194519714231575