| L(s) = 1 | + 2·3-s + 4·7-s + 9-s + 4·13-s + 4·19-s + 8·21-s − 6·23-s − 4·27-s − 6·29-s − 8·31-s + 2·37-s + 8·39-s − 6·41-s + 8·43-s + 6·47-s + 9·49-s − 6·53-s + 8·57-s − 12·59-s + 2·61-s + 4·63-s + 10·67-s − 12·69-s + 12·71-s − 16·73-s + 8·79-s − 11·81-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 1/3·9-s + 1.10·13-s + 0.917·19-s + 1.74·21-s − 1.25·23-s − 0.769·27-s − 1.11·29-s − 1.43·31-s + 0.328·37-s + 1.28·39-s − 0.937·41-s + 1.21·43-s + 0.875·47-s + 9/7·49-s − 0.824·53-s + 1.05·57-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + 1.22·67-s − 1.44·69-s + 1.42·71-s − 1.87·73-s + 0.900·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 193600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−13.56025356542833, −12.97536777183197, −12.39433602303451, −11.90345230478932, −11.31666416564915, −11.03502113035927, −10.67406872751058, −9.892909567677809, −9.313737015350381, −9.110917474431207, −8.462561819519349, −8.131767556429586, −7.636034497559044, −7.481570001106743, −6.646786227159146, −5.906588220937866, −5.526212318537752, −5.067047327775435, −4.218984976280917, −3.885220357713478, −3.436154215551848, −2.714735556415924, −2.064894744672264, −1.658558737295685, −1.124139622344984, 0,
1.124139622344984, 1.658558737295685, 2.064894744672264, 2.714735556415924, 3.436154215551848, 3.885220357713478, 4.218984976280917, 5.067047327775435, 5.526212318537752, 5.906588220937866, 6.646786227159146, 7.481570001106743, 7.636034497559044, 8.131767556429586, 8.462561819519349, 9.110917474431207, 9.313737015350381, 9.892909567677809, 10.67406872751058, 11.03502113035927, 11.31666416564915, 11.90345230478932, 12.39433602303451, 12.97536777183197, 13.56025356542833
