| L(s) = 1 | − 3·3-s + 5·5-s − 24.8·7-s + 9·9-s + 25.7·11-s + 60.6·13-s − 15·15-s − 28.6·17-s − 86.6·19-s + 74.6·21-s + 52.3·23-s + 25·25-s − 27·27-s + 6·29-s − 84.8·31-s − 77.2·33-s − 124.·35-s − 448.·37-s − 181.·39-s + 183.·41-s − 252·43-s + 45·45-s − 41.9·47-s + 275.·49-s + 85.8·51-s − 228.·53-s + 128.·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.34·7-s + 0.333·9-s + 0.705·11-s + 1.29·13-s − 0.258·15-s − 0.408·17-s − 1.04·19-s + 0.775·21-s + 0.474·23-s + 0.200·25-s − 0.192·27-s + 0.0384·29-s − 0.491·31-s − 0.407·33-s − 0.600·35-s − 1.99·37-s − 0.746·39-s + 0.697·41-s − 0.893·43-s + 0.149·45-s − 0.130·47-s + 0.802·49-s + 0.235·51-s − 0.592·53-s + 0.315·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 480 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 24.8T + 343T^{2} \) |
| 11 | \( 1 - 25.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 60.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 28.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 86.6T + 6.85e3T^{2} \) |
| 23 | \( 1 - 52.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 6T + 2.43e4T^{2} \) |
| 31 | \( 1 + 84.8T + 2.97e4T^{2} \) |
| 37 | \( 1 + 448.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 183.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 252T + 7.95e4T^{2} \) |
| 47 | \( 1 + 41.9T + 1.03e5T^{2} \) |
| 53 | \( 1 + 228.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 179.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 480.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 855.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 675.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 621.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 513.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.28e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.00e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 300.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.24206236017421324497971779994, −9.237293780580162233252156303739, −8.601770316464831612532068220452, −6.94881621638322817815915435590, −6.41950423694611921858344852694, −5.64831386449358235648814607396, −4.22129904073145964517593035058, −3.20401236164088282499934102484, −1.55466867004779837561600032350, 0,
1.55466867004779837561600032350, 3.20401236164088282499934102484, 4.22129904073145964517593035058, 5.64831386449358235648814607396, 6.41950423694611921858344852694, 6.94881621638322817815915435590, 8.601770316464831612532068220452, 9.237293780580162233252156303739, 10.24206236017421324497971779994
