| L(s) = 1 | − 3·3-s − 5-s + 6·9-s + 2·11-s + 6·13-s + 3·15-s + 9·23-s + 25-s − 9·27-s − 3·29-s + 2·31-s − 6·33-s − 8·37-s − 18·39-s + 5·41-s + 43-s − 6·45-s − 8·47-s + 4·53-s − 2·55-s + 8·59-s + 7·61-s − 6·65-s − 3·67-s − 27·69-s − 8·71-s + 14·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.447·5-s + 2·9-s + 0.603·11-s + 1.66·13-s + 0.774·15-s + 1.87·23-s + 1/5·25-s − 1.73·27-s − 0.557·29-s + 0.359·31-s − 1.04·33-s − 1.31·37-s − 2.88·39-s + 0.780·41-s + 0.152·43-s − 0.894·45-s − 1.16·47-s + 0.549·53-s − 0.269·55-s + 1.04·59-s + 0.896·61-s − 0.744·65-s − 0.366·67-s − 3.25·69-s − 0.949·71-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 283220 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.84162666495983, −12.44240413550249, −11.95282378432447, −11.40127972159569, −11.23772988536399, −10.88226797554140, −10.45666183332835, −9.853981633672849, −9.358574576849488, −8.684984352735991, −8.500704876719286, −7.722109116192253, −7.028358325190069, −6.851310235927311, −6.404188092206793, −5.802497863569207, −5.461266766595094, −4.942034266909353, −4.423275869994462, −3.828463876980318, −3.509984452281575, −2.733609028682295, −1.727808115411611, −1.165924731533219, −0.8305694428198393, 0,
0.8305694428198393, 1.165924731533219, 1.727808115411611, 2.733609028682295, 3.509984452281575, 3.828463876980318, 4.423275869994462, 4.942034266909353, 5.461266766595094, 5.802497863569207, 6.404188092206793, 6.851310235927311, 7.028358325190069, 7.722109116192253, 8.500704876719286, 8.684984352735991, 9.358574576849488, 9.853981633672849, 10.45666183332835, 10.88226797554140, 11.23772988536399, 11.40127972159569, 11.95282378432447, 12.44240413550249, 12.84162666495983
