| L(s) = 1 | − 2-s − 4-s + 3·5-s + 3·8-s − 3·10-s + 3·11-s − 16-s − 2·17-s + 19-s − 3·20-s − 3·22-s + 4·25-s − 7·29-s − 3·31-s − 5·32-s + 2·34-s + 2·37-s − 38-s + 9·40-s + 3·41-s − 7·43-s − 3·44-s + 47-s − 4·50-s − 3·53-s + 9·55-s + 7·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.34·5-s + 1.06·8-s − 0.948·10-s + 0.904·11-s − 1/4·16-s − 0.485·17-s + 0.229·19-s − 0.670·20-s − 0.639·22-s + 4/5·25-s − 1.29·29-s − 0.538·31-s − 0.883·32-s + 0.342·34-s + 0.328·37-s − 0.162·38-s + 1.42·40-s + 0.468·41-s − 1.06·43-s − 0.452·44-s + 0.145·47-s − 0.565·50-s − 0.412·53-s + 1.21·55-s + 0.919·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 7 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 5 T + p T^{2} \) |
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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
−14.23238376168294, −13.79417447465971, −13.33126814844038, −13.01922098030834, −12.44667418791616, −11.72010599378315, −11.14129970499049, −10.74718167036576, −10.01086133129452, −9.764989908558345, −9.248106812159396, −8.946574376746650, −8.421390230449031, −7.682843944885076, −7.227704694206517, −6.515670496952020, −6.125565109713143, −5.345348955164088, −5.096067319322527, −4.179220269914585, −3.810542327030979, −2.917902096142587, −2.029677990190593, −1.660394011711048, −0.9686313601887893, 0,
0.9686313601887893, 1.660394011711048, 2.029677990190593, 2.917902096142587, 3.810542327030979, 4.179220269914585, 5.096067319322527, 5.345348955164088, 6.125565109713143, 6.515670496952020, 7.227704694206517, 7.682843944885076, 8.421390230449031, 8.946574376746650, 9.248106812159396, 9.764989908558345, 10.01086133129452, 10.74718167036576, 11.14129970499049, 11.72010599378315, 12.44667418791616, 13.01922098030834, 13.33126814844038, 13.79417447465971, 14.23238376168294
