| L(s) = 1 | − 5.17·3-s − 10.0·5-s + 7·7-s − 0.259·9-s − 11·11-s − 84.5·13-s + 52.1·15-s − 38.2·17-s + 127.·19-s − 36.1·21-s − 140.·23-s − 23.3·25-s + 140.·27-s − 116.·29-s − 338.·31-s + 56.8·33-s − 70.5·35-s − 75.3·37-s + 437.·39-s − 22.4·41-s − 181.·43-s + 2.61·45-s − 300.·47-s + 49·49-s + 197.·51-s − 31.8·53-s + 110.·55-s + ⋯ |
| L(s) = 1 | − 0.995·3-s − 0.901·5-s + 0.377·7-s − 0.00960·9-s − 0.301·11-s − 1.80·13-s + 0.897·15-s − 0.545·17-s + 1.53·19-s − 0.376·21-s − 1.27·23-s − 0.186·25-s + 1.00·27-s − 0.747·29-s − 1.96·31-s + 0.300·33-s − 0.340·35-s − 0.334·37-s + 1.79·39-s − 0.0854·41-s − 0.644·43-s + 0.00865·45-s − 0.932·47-s + 0.142·49-s + 0.543·51-s − 0.0825·53-s + 0.271·55-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
| \(L(2)\) |
\(\approx\) |
\(0.1893226814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1893226814\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.17T + 27T^{2} \) |
| 5 | \( 1 + 10.0T + 125T^{2} \) |
| 13 | \( 1 + 84.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 38.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 140.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 116.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 338.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 75.3T + 5.06e4T^{2} \) |
| 41 | \( 1 + 22.4T + 6.89e4T^{2} \) |
| 43 | \( 1 + 181.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 300.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 31.8T + 1.48e5T^{2} \) |
| 59 | \( 1 - 68.3T + 2.05e5T^{2} \) |
| 61 | \( 1 + 145.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 668.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 416.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 458.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 355.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 935.T + 9.12e5T^{2} \) |
| 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
−9.510395971316266183628219458135, −8.374612613641207276614766146335, −7.50077024031823612106992049921, −7.09601242843573675554713544111, −5.74798232953495470437783446747, −5.18772456039638925425717161194, −4.35433900103875526129779816371, −3.21678272456601636832384188498, −1.90720178597180734805306218556, −0.22214449369852328533142672289,
0.22214449369852328533142672289, 1.90720178597180734805306218556, 3.21678272456601636832384188498, 4.35433900103875526129779816371, 5.18772456039638925425717161194, 5.74798232953495470437783446747, 7.09601242843573675554713544111, 7.50077024031823612106992049921, 8.374612613641207276614766146335, 9.510395971316266183628219458135
