| L(s) = 1 | + 2.19·3-s − 1.70·5-s − 7-s + 1.81·9-s + 4.56·11-s − 13-s − 3.74·15-s + 7.37·17-s + 6.46·19-s − 2.19·21-s + 5.90·23-s − 2.08·25-s − 2.59·27-s + 2.69·29-s + 0.523·31-s + 10.0·33-s + 1.70·35-s + 2.17·37-s − 2.19·39-s − 4.98·41-s + 2.32·43-s − 3.10·45-s − 12.3·47-s + 49-s + 16.1·51-s − 4.71·53-s − 7.78·55-s + ⋯ |
| L(s) = 1 | + 1.26·3-s − 0.763·5-s − 0.377·7-s + 0.605·9-s + 1.37·11-s − 0.277·13-s − 0.967·15-s + 1.78·17-s + 1.48·19-s − 0.478·21-s + 1.23·23-s − 0.417·25-s − 0.500·27-s + 0.500·29-s + 0.0939·31-s + 1.74·33-s + 0.288·35-s + 0.357·37-s − 0.351·39-s − 0.778·41-s + 0.355·43-s − 0.462·45-s − 1.79·47-s + 0.142·49-s + 2.26·51-s − 0.648·53-s − 1.05·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159193416\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159193416\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.19T + 3T^{2} \) |
| 5 | \( 1 + 1.70T + 5T^{2} \) |
| 11 | \( 1 - 4.56T + 11T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 6.46T + 19T^{2} \) |
| 23 | \( 1 - 5.90T + 23T^{2} \) |
| 29 | \( 1 - 2.69T + 29T^{2} \) |
| 31 | \( 1 - 0.523T + 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 + 4.98T + 41T^{2} \) |
| 43 | \( 1 - 2.32T + 43T^{2} \) |
| 47 | \( 1 + 12.3T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 + 13.7T + 59T^{2} \) |
| 61 | \( 1 + 0.137T + 61T^{2} \) |
| 67 | \( 1 + 7.96T + 67T^{2} \) |
| 71 | \( 1 + 13.8T + 71T^{2} \) |
| 73 | \( 1 - 2.52T + 73T^{2} \) |
| 79 | \( 1 - 4.85T + 79T^{2} \) |
| 83 | \( 1 + 7.51T + 83T^{2} \) |
| 89 | \( 1 + 10.8T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−10.02823668633485855492574632265, −9.451965357010081392210619076353, −8.723545732676100815400745459616, −7.75279397320532598205509283406, −7.29512002740419805324966312970, −6.04892275863511645120660970772, −4.71590911852461806532621101903, −3.38407850066359012084685799065, −3.21478602642920728913889105071, −1.33717339232557373012580302640,
1.33717339232557373012580302640, 3.21478602642920728913889105071, 3.38407850066359012084685799065, 4.71590911852461806532621101903, 6.04892275863511645120660970772, 7.29512002740419805324966312970, 7.75279397320532598205509283406, 8.723545732676100815400745459616, 9.451965357010081392210619076353, 10.02823668633485855492574632265
