| L(s) = 1 | + 0.502·3-s − 3.38·5-s − 3.39·7-s − 2.74·9-s + 0.762·11-s + 5.78·13-s − 1.70·15-s − 6.42·17-s − 2.41·19-s − 1.70·21-s − 3.05·23-s + 6.46·25-s − 2.88·27-s − 6.29·29-s + 2.62·31-s + 0.382·33-s + 11.5·35-s − 11.9·37-s + 2.90·39-s + 1.26·41-s − 12.2·43-s + 9.30·45-s − 4.26·47-s + 4.54·49-s − 3.22·51-s + 3.63·53-s − 2.58·55-s + ⋯ |
| L(s) = 1 | + 0.289·3-s − 1.51·5-s − 1.28·7-s − 0.915·9-s + 0.229·11-s + 1.60·13-s − 0.439·15-s − 1.55·17-s − 0.554·19-s − 0.372·21-s − 0.636·23-s + 1.29·25-s − 0.555·27-s − 1.16·29-s + 0.472·31-s + 0.0666·33-s + 1.94·35-s − 1.96·37-s + 0.465·39-s + 0.197·41-s − 1.86·43-s + 1.38·45-s − 0.622·47-s + 0.648·49-s − 0.451·51-s + 0.499·53-s − 0.348·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2418276384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2418276384\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.502T + 3T^{2} \) |
| 5 | \( 1 + 3.38T + 5T^{2} \) |
| 7 | \( 1 + 3.39T + 7T^{2} \) |
| 11 | \( 1 - 0.762T + 11T^{2} \) |
| 13 | \( 1 - 5.78T + 13T^{2} \) |
| 17 | \( 1 + 6.42T + 17T^{2} \) |
| 19 | \( 1 + 2.41T + 19T^{2} \) |
| 23 | \( 1 + 3.05T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 2.62T + 31T^{2} \) |
| 37 | \( 1 + 11.9T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 4.26T + 47T^{2} \) |
| 53 | \( 1 - 3.63T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 0.489T + 67T^{2} \) |
| 71 | \( 1 + 1.37T + 71T^{2} \) |
| 73 | \( 1 - 9.41T + 73T^{2} \) |
| 79 | \( 1 - 4.52T + 79T^{2} \) |
| 83 | \( 1 - 8.13T + 83T^{2} \) |
| 89 | \( 1 + 8.29T + 89T^{2} \) |
| 97 | \( 1 + 16.6T + 97T^{2} \) |
| 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
−8.050998898815550532109528931524, −7.02584642775016640830973458795, −6.52279899193720986378383356187, −5.98519496476669289989441540319, −4.91212647811453564998221209643, −3.85208759385569084619769382856, −3.68366473163545822819087051702, −2.96813823295025252551484355604, −1.82475575030966916162842426785, −0.22659818231969218225930656124,
0.22659818231969218225930656124, 1.82475575030966916162842426785, 2.96813823295025252551484355604, 3.68366473163545822819087051702, 3.85208759385569084619769382856, 4.91212647811453564998221209643, 5.98519496476669289989441540319, 6.52279899193720986378383356187, 7.02584642775016640830973458795, 8.050998898815550532109528931524
