| L(s) = 1 | − 0.414·2-s − 1.82·4-s + 0.585·5-s − 4.82·7-s + 1.58·8-s − 0.242·10-s − 11-s − 13-s + 1.99·14-s + 3·16-s + 5.41·17-s − 6·19-s − 1.07·20-s + 0.414·22-s + 0.828·23-s − 4.65·25-s + 0.414·26-s + 8.82·28-s − 4.24·29-s + 4.24·31-s − 4.41·32-s − 2.24·34-s − 2.82·35-s + 7.65·37-s + 2.48·38-s + 0.928·40-s + 12·41-s + ⋯ |
| L(s) = 1 | − 0.292·2-s − 0.914·4-s + 0.261·5-s − 1.82·7-s + 0.560·8-s − 0.0767·10-s − 0.301·11-s − 0.277·13-s + 0.534·14-s + 0.750·16-s + 1.31·17-s − 1.37·19-s − 0.239·20-s + 0.0883·22-s + 0.172·23-s − 0.931·25-s + 0.0812·26-s + 1.66·28-s − 0.787·29-s + 0.762·31-s − 0.780·32-s − 0.384·34-s − 0.478·35-s + 1.25·37-s + 0.403·38-s + 0.146·40-s + 1.87·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7284988757\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7284988757\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| good | 2 | \( 1 + 0.414T + 2T^{2} \) |
| 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 17 | \( 1 - 5.41T + 17T^{2} \) |
| 19 | \( 1 + 6T + 19T^{2} \) |
| 23 | \( 1 - 0.828T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4.24T + 31T^{2} \) |
| 37 | \( 1 - 7.65T + 37T^{2} \) |
| 41 | \( 1 - 12T + 41T^{2} \) |
| 43 | \( 1 - 5.07T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 13.3T + 53T^{2} \) |
| 59 | \( 1 - 10.8T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 + 3.07T + 67T^{2} \) |
| 71 | \( 1 - 5.65T + 71T^{2} \) |
| 73 | \( 1 - 12.4T + 73T^{2} \) |
| 79 | \( 1 + 9.07T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| 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
−9.625099179804185306286044665132, −9.146717418908801290796688895248, −8.074417467551985776467335860107, −7.34992894752340496567865151341, −6.15766295930498077119774650062, −5.70617041445339826377609649001, −4.38643152812328131981952726633, −3.58625180362604274215925081187, −2.50433672999441811469743309360, −0.63986317949166311405880976163,
0.63986317949166311405880976163, 2.50433672999441811469743309360, 3.58625180362604274215925081187, 4.38643152812328131981952726633, 5.70617041445339826377609649001, 6.15766295930498077119774650062, 7.34992894752340496567865151341, 8.074417467551985776467335860107, 9.146717418908801290796688895248, 9.625099179804185306286044665132
