| L(s) = 1 | + 3·3-s + 4·5-s + 7-s + 6·9-s + 5·13-s + 12·15-s − 5·17-s + 19-s + 3·21-s − 3·23-s + 11·25-s + 9·27-s − 7·29-s − 10·31-s + 4·35-s + 2·37-s + 15·39-s + 6·41-s + 4·43-s + 24·45-s − 8·47-s − 6·49-s − 15·51-s + 9·53-s + 3·57-s − 59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 1.78·5-s + 0.377·7-s + 2·9-s + 1.38·13-s + 3.09·15-s − 1.21·17-s + 0.229·19-s + 0.654·21-s − 0.625·23-s + 11/5·25-s + 1.73·27-s − 1.29·29-s − 1.79·31-s + 0.676·35-s + 0.328·37-s + 2.40·39-s + 0.937·41-s + 0.609·43-s + 3.57·45-s − 1.16·47-s − 6/7·49-s − 2.10·51-s + 1.23·53-s + 0.397·57-s − 0.130·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.807306712\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.807306712\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 7 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.516953861454534288839268534455, −7.67400993227461590838049486915, −6.91811254047735942143181299092, −6.08169850106264326305196868557, −5.47273971444144910988344490699, −4.34064751029975278777807269437, −3.61817507134350571960281466872, −2.67838219974142762085046315525, −1.92977995463815202835161772847, −1.50372834764025438067135938320,
1.50372834764025438067135938320, 1.92977995463815202835161772847, 2.67838219974142762085046315525, 3.61817507134350571960281466872, 4.34064751029975278777807269437, 5.47273971444144910988344490699, 6.08169850106264326305196868557, 6.91811254047735942143181299092, 7.67400993227461590838049486915, 8.516953861454534288839268534455
