| L(s) = 1 | + 3.40·2-s − 3·3-s + 3.58·4-s + 5.47·5-s − 10.2·6-s + 7·7-s − 15.0·8-s + 9·9-s + 18.6·10-s + 43.1·11-s − 10.7·12-s + 7.84·13-s + 23.8·14-s − 16.4·15-s − 79.8·16-s − 55.9·17-s + 30.6·18-s + 122.·19-s + 19.6·20-s − 21·21-s + 146.·22-s + 23·23-s + 45.0·24-s − 94.9·25-s + 26.6·26-s − 27·27-s + 25.1·28-s + ⋯ |
| L(s) = 1 | + 1.20·2-s − 0.577·3-s + 0.448·4-s + 0.490·5-s − 0.694·6-s + 0.377·7-s − 0.663·8-s + 0.333·9-s + 0.589·10-s + 1.18·11-s − 0.259·12-s + 0.167·13-s + 0.454·14-s − 0.282·15-s − 1.24·16-s − 0.798·17-s + 0.401·18-s + 1.48·19-s + 0.219·20-s − 0.218·21-s + 1.42·22-s + 0.208·23-s + 0.383·24-s − 0.759·25-s + 0.201·26-s − 0.192·27-s + 0.169·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.414892172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.414892172\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 - 23T \) |
| good | 2 | \( 1 - 3.40T + 8T^{2} \) |
| 5 | \( 1 - 5.47T + 125T^{2} \) |
| 11 | \( 1 - 43.1T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.84T + 2.19e3T^{2} \) |
| 17 | \( 1 + 55.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 29 | \( 1 - 199.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 119.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 303.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 380.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 415.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 365.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 515.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 348.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 226.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 724.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 205.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.35e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 691.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 448.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 871.T + 9.12e5T^{2} \) |
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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
−10.93591469089812332970254216065, −9.621888192603019994425717307408, −9.015932437956289552515384942212, −7.56160786681170090538663179701, −6.35033971816629752686387458854, −5.88324598991086901663518082260, −4.75193421878209597587132686343, −4.06963975785877403503555128508, −2.69276811688849924500056732230, −1.07271769591559464084527924167,
1.07271769591559464084527924167, 2.69276811688849924500056732230, 4.06963975785877403503555128508, 4.75193421878209597587132686343, 5.88324598991086901663518082260, 6.35033971816629752686387458854, 7.56160786681170090538663179701, 9.015932437956289552515384942212, 9.621888192603019994425717307408, 10.93591469089812332970254216065
