| L(s) = 1 | + 4.57·2-s + 7.79·3-s + 12.8·4-s + 13.2·5-s + 35.6·6-s − 25.2·7-s + 22.3·8-s + 33.8·9-s + 60.7·10-s + 58.0·11-s + 100.·12-s + 44.0·13-s − 115.·14-s + 103.·15-s − 0.999·16-s − 17·17-s + 154.·18-s − 56.7·19-s + 171.·20-s − 196.·21-s + 265.·22-s − 19.4·23-s + 174.·24-s + 51.8·25-s + 201.·26-s + 53.2·27-s − 324.·28-s + ⋯ |
| L(s) = 1 | + 1.61·2-s + 1.50·3-s + 1.61·4-s + 1.18·5-s + 2.42·6-s − 1.36·7-s + 0.987·8-s + 1.25·9-s + 1.92·10-s + 1.59·11-s + 2.41·12-s + 0.940·13-s − 2.19·14-s + 1.78·15-s − 0.0156·16-s − 0.242·17-s + 2.02·18-s − 0.685·19-s + 1.91·20-s − 2.04·21-s + 2.57·22-s − 0.175·23-s + 1.48·24-s + 0.415·25-s + 1.51·26-s + 0.379·27-s − 2.19·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 731 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.879403913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.879403913\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 17 | \( 1 + 17T \) |
| 43 | \( 1 + 43T \) |
| good | 2 | \( 1 - 4.57T + 8T^{2} \) |
| 3 | \( 1 - 7.79T + 27T^{2} \) |
| 5 | \( 1 - 13.2T + 125T^{2} \) |
| 7 | \( 1 + 25.2T + 343T^{2} \) |
| 11 | \( 1 - 58.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 44.0T + 2.19e3T^{2} \) |
| 19 | \( 1 + 56.7T + 6.85e3T^{2} \) |
| 23 | \( 1 + 19.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 180.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 86.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 56.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 77.0T + 6.89e4T^{2} \) |
| 47 | \( 1 + 27.2T + 1.03e5T^{2} \) |
| 53 | \( 1 + 62.3T + 1.48e5T^{2} \) |
| 59 | \( 1 - 190.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 690.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 217.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 607.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 202.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 373.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 364.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 809.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.55e3T + 9.12e5T^{2} \) |
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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
−9.642993176953228344292407798693, −9.339157175892690261683921335529, −8.428342654079716082098910250021, −6.78380151709986237658483717270, −6.46296684976616970683833259153, −5.57935649780354211906505581230, −3.97823187638316072765859654674, −3.63311074799726927077517399816, −2.60925398703790738816022620284, −1.72122449918906614131905988910,
1.72122449918906614131905988910, 2.60925398703790738816022620284, 3.63311074799726927077517399816, 3.97823187638316072765859654674, 5.57935649780354211906505581230, 6.46296684976616970683833259153, 6.78380151709986237658483717270, 8.428342654079716082098910250021, 9.339157175892690261683921335529, 9.642993176953228344292407798693
