| L(s) = 1 | + 1.40·2-s − 4.22·3-s − 6.03·4-s + 13.7·5-s − 5.91·6-s + 7·7-s − 19.6·8-s − 9.15·9-s + 19.2·10-s + 43.5·11-s + 25.5·12-s − 9.99·13-s + 9.80·14-s − 58.1·15-s + 20.7·16-s + 113.·17-s − 12.8·18-s + 118.·19-s − 83.1·20-s − 29.5·21-s + 61.0·22-s + 23·23-s + 83.0·24-s + 64.7·25-s − 14.0·26-s + 152.·27-s − 42.2·28-s + ⋯ |
| L(s) = 1 | + 0.495·2-s − 0.813·3-s − 0.754·4-s + 1.23·5-s − 0.402·6-s + 0.377·7-s − 0.868·8-s − 0.338·9-s + 0.610·10-s + 1.19·11-s + 0.613·12-s − 0.213·13-s + 0.187·14-s − 1.00·15-s + 0.324·16-s + 1.62·17-s − 0.167·18-s + 1.42·19-s − 0.930·20-s − 0.307·21-s + 0.591·22-s + 0.208·23-s + 0.706·24-s + 0.518·25-s − 0.105·26-s + 1.08·27-s − 0.285·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 161 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.766135598\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.766135598\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 1.40T + 8T^{2} \) |
| 3 | \( 1 + 4.22T + 27T^{2} \) |
| 5 | \( 1 - 13.7T + 125T^{2} \) |
| 11 | \( 1 - 43.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 9.99T + 2.19e3T^{2} \) |
| 17 | \( 1 - 113.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 118.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 270.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 112.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 281.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 348.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 300.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 716.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 156.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 226.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 148.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 895.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 271.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 146.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.21e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 305.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.65e3T + 9.12e5T^{2} \) |
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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
−12.17261667793432128146247624667, −11.85349936033971074989646534970, −10.26601642222338454386138322876, −9.536860636487478975992414696430, −8.460166329130268297949575399239, −6.68116816522779527540533487343, −5.56370738833505041045462936548, −5.09612098995609420887946885001, −3.34385113671348148932638683084, −1.14689749614727222328166067807,
1.14689749614727222328166067807, 3.34385113671348148932638683084, 5.09612098995609420887946885001, 5.56370738833505041045462936548, 6.68116816522779527540533487343, 8.460166329130268297949575399239, 9.536860636487478975992414696430, 10.26601642222338454386138322876, 11.85349936033971074989646534970, 12.17261667793432128146247624667
