| L(s) = 1 | − 2·3-s + 2·5-s − 7-s + 9-s + 11-s − 4·15-s + 4·17-s − 4·19-s + 2·21-s + 4·23-s − 25-s + 4·27-s + 2·29-s + 2·31-s − 2·33-s − 2·35-s − 6·37-s + 4·41-s + 4·43-s + 2·45-s − 2·47-s + 49-s − 8·51-s + 2·53-s + 2·55-s + 8·57-s + 6·59-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 1.03·15-s + 0.970·17-s − 0.917·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 0.371·29-s + 0.359·31-s − 0.348·33-s − 0.338·35-s − 0.986·37-s + 0.624·41-s + 0.609·43-s + 0.298·45-s − 0.291·47-s + 1/7·49-s − 1.12·51-s + 0.274·53-s + 0.269·55-s + 1.05·57-s + 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.191988767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.191988767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−9.845744954415415818804338785295, −9.093426413870990530848586861432, −8.100815479365426802534454930602, −6.87284055376711985333769557969, −6.30774181798334847916241962474, −5.58128014559474822553080706315, −4.89445170768043674449756720974, −3.61305926026132330506898704685, −2.29949696282704383166659510479, −0.870787299639210311104118483147,
0.870787299639210311104118483147, 2.29949696282704383166659510479, 3.61305926026132330506898704685, 4.89445170768043674449756720974, 5.58128014559474822553080706315, 6.30774181798334847916241962474, 6.87284055376711985333769557969, 8.100815479365426802534454930602, 9.093426413870990530848586861432, 9.845744954415415818804338785295
