L(s) = 1 | + 2·5-s + 2·7-s − 4·13-s + 2·17-s + 4·19-s + 4·23-s − 25-s + 6·29-s + 2·31-s + 4·35-s − 8·37-s + 2·41-s + 4·43-s + 12·47-s − 3·49-s + 6·53-s + 4·59-s − 8·65-s − 12·67-s + 12·71-s + 6·73-s + 10·79-s − 16·83-s + 4·85-s − 10·89-s − 8·91-s + 8·95-s + ⋯ |
L(s) = 1 | + 0.894·5-s + 0.755·7-s − 1.10·13-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s + 1.11·29-s + 0.359·31-s + 0.676·35-s − 1.31·37-s + 0.312·41-s + 0.609·43-s + 1.75·47-s − 3/7·49-s + 0.824·53-s + 0.520·59-s − 0.992·65-s − 1.46·67-s + 1.42·71-s + 0.702·73-s + 1.12·79-s − 1.75·83-s + 0.433·85-s − 1.05·89-s − 0.838·91-s + 0.820·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.373927419\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.373927419\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 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
−9.098740161279956346119573900465, −8.234914851164955606889524890008, −7.42769147265230109024706401485, −6.78042831829562695928275409894, −5.63263291479472794167609115304, −5.20906307637637876459942459972, −4.31817750759109547068712590842, −3.02006258632911954070275945877, −2.16659637218800472387918779639, −1.06061452190242472304939992110,
1.06061452190242472304939992110, 2.16659637218800472387918779639, 3.02006258632911954070275945877, 4.31817750759109547068712590842, 5.20906307637637876459942459972, 5.63263291479472794167609115304, 6.78042831829562695928275409894, 7.42769147265230109024706401485, 8.234914851164955606889524890008, 9.098740161279956346119573900465