L(s) = 1 | + 7.70·3-s + 5.23·5-s + 7·7-s + 32.4·9-s + 11·11-s − 29.3·13-s + 40.3·15-s + 107.·17-s − 34.2·19-s + 53.9·21-s − 10.3·23-s − 97.5·25-s + 41.7·27-s + 268.·29-s + 300.·31-s + 84.7·33-s + 36.6·35-s − 213.·37-s − 226.·39-s + 481.·41-s − 447.·43-s + 169.·45-s + 335.·47-s + 49·49-s + 830.·51-s + 183.·53-s + 57.5·55-s + ⋯ |
L(s) = 1 | + 1.48·3-s + 0.468·5-s + 0.377·7-s + 1.20·9-s + 0.301·11-s − 0.626·13-s + 0.694·15-s + 1.53·17-s − 0.414·19-s + 0.560·21-s − 0.0941·23-s − 0.780·25-s + 0.297·27-s + 1.72·29-s + 1.73·31-s + 0.447·33-s + 0.177·35-s − 0.947·37-s − 0.928·39-s + 1.83·41-s − 1.58·43-s + 0.562·45-s + 1.04·47-s + 0.142·49-s + 2.28·51-s + 0.476·53-s + 0.141·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.653724345\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.653724345\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| 11 | \( 1 - 11T \) |
good | 3 | \( 1 - 7.70T + 27T^{2} \) |
| 5 | \( 1 - 5.23T + 125T^{2} \) |
| 13 | \( 1 + 29.3T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 34.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 10.3T + 1.21e4T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 213.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 481.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 447.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 335.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 183.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 58.1T + 2.05e5T^{2} \) |
| 61 | \( 1 + 624.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 226.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.12e3T + 3.57e5T^{2} \) |
| 73 | \( 1 + 87.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 485.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 884.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.28e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 49.2T + 9.12e5T^{2} \) |
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
−9.401989796773484493786141131589, −8.337832938183471724111819240333, −8.034646372460423703545050656148, −7.07696976599093434256655544665, −6.05416774632017833803569543200, −4.95415330407543640987994503600, −3.96578346283810094851852650541, −2.96874686061053410119953447341, −2.21449998889540270345320816908, −1.09856246365125948909741431018,
1.09856246365125948909741431018, 2.21449998889540270345320816908, 2.96874686061053410119953447341, 3.96578346283810094851852650541, 4.95415330407543640987994503600, 6.05416774632017833803569543200, 7.07696976599093434256655544665, 8.034646372460423703545050656148, 8.337832938183471724111819240333, 9.401989796773484493786141131589