| L(s) = 1 | + 3-s − 2·7-s − 2·9-s − 13-s + 4·17-s + 4·19-s − 2·21-s + 23-s − 5·27-s − 3·29-s + 31-s + 8·37-s − 39-s − 5·41-s − 6·43-s + 9·47-s − 3·49-s + 4·51-s − 2·53-s + 4·57-s + 4·63-s + 4·67-s + 69-s − 3·71-s − 7·73-s − 4·79-s + 81-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s − 2/3·9-s − 0.277·13-s + 0.970·17-s + 0.917·19-s − 0.436·21-s + 0.208·23-s − 0.962·27-s − 0.557·29-s + 0.179·31-s + 1.31·37-s − 0.160·39-s − 0.780·41-s − 0.914·43-s + 1.31·47-s − 3/7·49-s + 0.560·51-s − 0.274·53-s + 0.529·57-s + 0.503·63-s + 0.488·67-s + 0.120·69-s − 0.356·71-s − 0.819·73-s − 0.450·79-s + 1/9·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.930800459\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.930800459\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.70331446220302113227359706670, −7.20308394206193869602930106291, −6.30036380396411373236224396287, −5.69707178249393477369390293267, −5.05536725543986125075521185626, −4.03719572981820868804390139578, −3.20393046177938685639495386167, −2.90782336309676735771563030634, −1.83941746391344063233010883703, −0.63526749850772052539174452888,
0.63526749850772052539174452888, 1.83941746391344063233010883703, 2.90782336309676735771563030634, 3.20393046177938685639495386167, 4.03719572981820868804390139578, 5.05536725543986125075521185626, 5.69707178249393477369390293267, 6.30036380396411373236224396287, 7.20308394206193869602930106291, 7.70331446220302113227359706670
