L(s) = 1 | + 3-s − 5-s + 7-s + 9-s + 5.12·11-s − 3.12·13-s − 15-s + 2·17-s − 1.12·19-s + 21-s + 25-s + 27-s − 2·29-s + 5.12·31-s + 5.12·33-s − 35-s − 2·37-s − 3.12·39-s + 2·41-s + 10.2·43-s − 45-s + 49-s + 2·51-s + 13.3·53-s − 5.12·55-s − 1.12·57-s − 4·59-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.377·7-s + 0.333·9-s + 1.54·11-s − 0.866·13-s − 0.258·15-s + 0.485·17-s − 0.257·19-s + 0.218·21-s + 0.200·25-s + 0.192·27-s − 0.371·29-s + 0.920·31-s + 0.891·33-s − 0.169·35-s − 0.328·37-s − 0.500·39-s + 0.312·41-s + 1.56·43-s − 0.149·45-s + 0.142·49-s + 0.280·51-s + 1.83·53-s − 0.690·55-s − 0.148·57-s − 0.520·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.456486751\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.456486751\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 3.12T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 5.12T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 + 2.24T + 67T^{2} \) |
| 71 | \( 1 + 5.12T + 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 2.24T + 79T^{2} \) |
| 83 | \( 1 + 4T + 83T^{2} \) |
| 89 | \( 1 + 0.246T + 89T^{2} \) |
| 97 | \( 1 - 4.87T + 97T^{2} \) |
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
−8.683111298661552996166599051147, −7.83650272054937963197442077868, −7.26810375070652902754854621751, −6.52644521261628989806269976064, −5.59059965720721089073493959986, −4.51152361870875064009721860461, −4.01974151868210735089249691941, −3.07337316899399977530952563071, −2.05679456993616796541029183111, −0.950001808002750825994859842931,
0.950001808002750825994859842931, 2.05679456993616796541029183111, 3.07337316899399977530952563071, 4.01974151868210735089249691941, 4.51152361870875064009721860461, 5.59059965720721089073493959986, 6.52644521261628989806269976064, 7.26810375070652902754854621751, 7.83650272054937963197442077868, 8.683111298661552996166599051147