| L(s) = 1 | + 3.29·3-s + 5-s − 1.54·7-s + 7.83·9-s − 3.54·11-s + 2.25·13-s + 3.29·15-s − 17-s + 4.83·19-s − 5.09·21-s + 1.54·23-s + 25-s + 15.9·27-s − 2.83·29-s − 7.87·31-s − 11.6·33-s − 1.54·35-s − 0.584·37-s + 7.42·39-s + 9.09·41-s + 2·43-s + 7.83·45-s − 6.83·47-s − 4.60·49-s − 3.29·51-s − 12.9·53-s − 3.54·55-s + ⋯ |
| L(s) = 1 | + 1.90·3-s + 0.447·5-s − 0.584·7-s + 2.61·9-s − 1.06·11-s + 0.625·13-s + 0.850·15-s − 0.242·17-s + 1.11·19-s − 1.11·21-s + 0.322·23-s + 0.200·25-s + 3.06·27-s − 0.527·29-s − 1.41·31-s − 2.03·33-s − 0.261·35-s − 0.0961·37-s + 1.18·39-s + 1.42·41-s + 0.304·43-s + 1.16·45-s − 0.997·47-s − 0.657·49-s − 0.461·51-s − 1.77·53-s − 0.478·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.857778637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.857778637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 3.29T + 3T^{2} \) |
| 7 | \( 1 + 1.54T + 7T^{2} \) |
| 11 | \( 1 + 3.54T + 11T^{2} \) |
| 13 | \( 1 - 2.25T + 13T^{2} \) |
| 19 | \( 1 - 4.83T + 19T^{2} \) |
| 23 | \( 1 - 1.54T + 23T^{2} \) |
| 29 | \( 1 + 2.83T + 29T^{2} \) |
| 31 | \( 1 + 7.87T + 31T^{2} \) |
| 37 | \( 1 + 0.584T + 37T^{2} \) |
| 41 | \( 1 - 9.09T + 41T^{2} \) |
| 43 | \( 1 - 2T + 43T^{2} \) |
| 47 | \( 1 + 6.83T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 3.74T + 61T^{2} \) |
| 67 | \( 1 + 10.2T + 67T^{2} \) |
| 71 | \( 1 + 4.78T + 71T^{2} \) |
| 73 | \( 1 + 6.83T + 73T^{2} \) |
| 79 | \( 1 - 2.64T + 79T^{2} \) |
| 83 | \( 1 - 2.51T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 11.7T + 97T^{2} \) |
| 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
−10.18316755096500608595004219485, −9.371808152746956672754849208271, −8.968770927909887985323401336851, −7.83885326553108985032820717044, −7.38394132900507575432246260437, −6.13119314931470421225192530411, −4.80998595787348960503927741885, −3.50029186207369379706280188496, −2.88395302779758627262914186463, −1.71637837809531664912056149507,
1.71637837809531664912056149507, 2.88395302779758627262914186463, 3.50029186207369379706280188496, 4.80998595787348960503927741885, 6.13119314931470421225192530411, 7.38394132900507575432246260437, 7.83885326553108985032820717044, 8.968770927909887985323401336851, 9.371808152746956672754849208271, 10.18316755096500608595004219485
