L(s) = 1 | + 2.96·3-s + 5-s + 1.02·7-s + 5.79·9-s + 0.504·13-s + 2.96·15-s + 3.93·17-s + 2.50·19-s + 3.05·21-s + 5.73·23-s + 25-s + 8.29·27-s − 6.92·29-s − 4.47·31-s + 1.02·35-s + 4.78·37-s + 1.49·39-s − 11.8·41-s − 7.10·43-s + 5.79·45-s + 0.182·47-s − 5.94·49-s + 11.6·51-s + 12.4·53-s + 7.42·57-s + 10.5·59-s + 10.9·61-s + ⋯ |
L(s) = 1 | + 1.71·3-s + 0.447·5-s + 0.388·7-s + 1.93·9-s + 0.139·13-s + 0.765·15-s + 0.953·17-s + 0.574·19-s + 0.665·21-s + 1.19·23-s + 0.200·25-s + 1.59·27-s − 1.28·29-s − 0.804·31-s + 0.173·35-s + 0.786·37-s + 0.239·39-s − 1.84·41-s − 1.08·43-s + 0.864·45-s + 0.0266·47-s − 0.848·49-s + 1.63·51-s + 1.71·53-s + 0.983·57-s + 1.37·59-s + 1.39·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.382709545\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.382709545\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
good | 3 | \( 1 - 2.96T + 3T^{2} \) |
| 7 | \( 1 - 1.02T + 7T^{2} \) |
| 13 | \( 1 - 0.504T + 13T^{2} \) |
| 17 | \( 1 - 3.93T + 17T^{2} \) |
| 19 | \( 1 - 2.50T + 19T^{2} \) |
| 23 | \( 1 - 5.73T + 23T^{2} \) |
| 29 | \( 1 + 6.92T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 7.10T + 43T^{2} \) |
| 47 | \( 1 - 0.182T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 10.5T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 9.76T + 67T^{2} \) |
| 71 | \( 1 - 13.9T + 71T^{2} \) |
| 73 | \( 1 - 7.32T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 4.33T + 83T^{2} \) |
| 89 | \( 1 - 17.2T + 89T^{2} \) |
| 97 | \( 1 + 7.80T + 97T^{2} \) |
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
−7.79008499556693129636641805537, −7.19880367345686873251913790494, −6.57564278031279631414157901536, −5.36439472364019774813948934281, −5.04280748222884461768269290548, −3.72063642787892067862517649448, −3.53418343708243384297911132340, −2.58302777551606828740840631937, −1.89427750446663924243953131400, −1.09884294597250210581477826037,
1.09884294597250210581477826037, 1.89427750446663924243953131400, 2.58302777551606828740840631937, 3.53418343708243384297911132340, 3.72063642787892067862517649448, 5.04280748222884461768269290548, 5.36439472364019774813948934281, 6.57564278031279631414157901536, 7.19880367345686873251913790494, 7.79008499556693129636641805537