| L(s) = 1 | + 3·3-s + 4.54·5-s − 7·7-s + 9·9-s − 40.7·11-s − 53.2·13-s + 13.6·15-s + 4.54·17-s + 122.·19-s − 21·21-s − 131.·23-s − 104.·25-s + 27·27-s + 216.·29-s + 251.·31-s − 122.·33-s − 31.8·35-s − 11.8·37-s − 159.·39-s − 111.·41-s + 369.·43-s + 40.9·45-s + 262.·47-s + 49·49-s + 13.6·51-s + 567.·53-s − 185.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.406·5-s − 0.377·7-s + 0.333·9-s − 1.11·11-s − 1.13·13-s + 0.234·15-s + 0.0649·17-s + 1.48·19-s − 0.218·21-s − 1.19·23-s − 0.834·25-s + 0.192·27-s + 1.38·29-s + 1.45·31-s − 0.644·33-s − 0.153·35-s − 0.0528·37-s − 0.656·39-s − 0.425·41-s + 1.30·43-s + 0.135·45-s + 0.815·47-s + 0.142·49-s + 0.0374·51-s + 1.46·53-s − 0.454·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1344 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.384915915\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.384915915\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 7 | \( 1 + 7T \) |
| good | 5 | \( 1 - 4.54T + 125T^{2} \) |
| 11 | \( 1 + 40.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 53.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 4.54T + 4.91e3T^{2} \) |
| 19 | \( 1 - 122.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 131.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 216.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 251.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 11.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 111.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 369.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 262.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 567.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 839.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 485.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 333.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 590.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 490.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 121.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 609.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 719.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 637.T + 9.12e5T^{2} \) |
| 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
−9.400492301415239746290306324714, −8.327276651260711185119573687609, −7.69918104545993394763569895851, −6.94642260521056048716137724396, −5.82981123944744025530687786941, −5.09162289729310630041079672206, −4.03738302754361848740736742301, −2.82214717362446157306408820133, −2.29488723284989378883591027340, −0.73234532025868511622353979724,
0.73234532025868511622353979724, 2.29488723284989378883591027340, 2.82214717362446157306408820133, 4.03738302754361848740736742301, 5.09162289729310630041079672206, 5.82981123944744025530687786941, 6.94642260521056048716137724396, 7.69918104545993394763569895851, 8.327276651260711185119573687609, 9.400492301415239746290306324714
