| L(s) = 1 | − 2.62·2-s − 0.0998·3-s + 4.90·4-s − 5-s + 0.262·6-s + 3.66·7-s − 7.63·8-s − 2.99·9-s + 2.62·10-s − 11-s − 0.489·12-s + 2.17·13-s − 9.63·14-s + 0.0998·15-s + 10.2·16-s + 17-s + 7.85·18-s + 0.404·19-s − 4.90·20-s − 0.366·21-s + 2.62·22-s + 7.06·23-s + 0.762·24-s + 25-s − 5.72·26-s + 0.598·27-s + 17.9·28-s + ⋯ |
| L(s) = 1 | − 1.85·2-s − 0.0576·3-s + 2.45·4-s − 0.447·5-s + 0.107·6-s + 1.38·7-s − 2.69·8-s − 0.996·9-s + 0.830·10-s − 0.301·11-s − 0.141·12-s + 0.603·13-s − 2.57·14-s + 0.0257·15-s + 2.56·16-s + 0.242·17-s + 1.85·18-s + 0.0927·19-s − 1.09·20-s − 0.0799·21-s + 0.560·22-s + 1.47·23-s + 0.155·24-s + 0.200·25-s − 1.12·26-s + 0.115·27-s + 3.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6476344311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6476344311\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 + 0.0998T + 3T^{2} \) |
| 7 | \( 1 - 3.66T + 7T^{2} \) |
| 13 | \( 1 - 2.17T + 13T^{2} \) |
| 19 | \( 1 - 0.404T + 19T^{2} \) |
| 23 | \( 1 - 7.06T + 23T^{2} \) |
| 29 | \( 1 + 3.97T + 29T^{2} \) |
| 31 | \( 1 + 1.26T + 31T^{2} \) |
| 37 | \( 1 - 6.03T + 37T^{2} \) |
| 41 | \( 1 + 6.95T + 41T^{2} \) |
| 43 | \( 1 + 4.47T + 43T^{2} \) |
| 47 | \( 1 - 6.76T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 - 2.51T + 61T^{2} \) |
| 67 | \( 1 + 4.53T + 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 12.1T + 73T^{2} \) |
| 79 | \( 1 - 0.825T + 79T^{2} \) |
| 83 | \( 1 - 1.63T + 83T^{2} \) |
| 89 | \( 1 + 2.44T + 89T^{2} \) |
| 97 | \( 1 - 9.80T + 97T^{2} \) |
| 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
−10.03387608940661930283153351239, −8.851518175783110947146992462191, −8.583982521332365456697433576786, −7.78432869653769882994997047566, −7.16537259781896361910418265882, −5.99085264804042262636804201552, −4.98567642179155058794925107993, −3.28248191612848222548303600965, −2.07509482612590405177789230191, −0.851918207362469217197482003618,
0.851918207362469217197482003618, 2.07509482612590405177789230191, 3.28248191612848222548303600965, 4.98567642179155058794925107993, 5.99085264804042262636804201552, 7.16537259781896361910418265882, 7.78432869653769882994997047566, 8.583982521332365456697433576786, 8.851518175783110947146992462191, 10.03387608940661930283153351239
