L(s) = 1 | − 0.869·3-s + 5-s − 1.85·7-s − 2.24·9-s − 5.49·11-s − 0.851·13-s − 0.869·15-s − 7.59·17-s − 6.07·19-s + 1.61·21-s − 3.96·23-s + 25-s + 4.56·27-s + 2.47·29-s − 3.46·31-s + 4.78·33-s − 1.85·35-s + 6.01·37-s + 0.740·39-s − 6.31·41-s − 1.43·43-s − 2.24·45-s − 10.4·47-s − 3.55·49-s + 6.60·51-s + 0.324·53-s − 5.49·55-s + ⋯ |
L(s) = 1 | − 0.502·3-s + 0.447·5-s − 0.701·7-s − 0.747·9-s − 1.65·11-s − 0.236·13-s − 0.224·15-s − 1.84·17-s − 1.39·19-s + 0.352·21-s − 0.827·23-s + 0.200·25-s + 0.877·27-s + 0.459·29-s − 0.621·31-s + 0.832·33-s − 0.313·35-s + 0.988·37-s + 0.118·39-s − 0.985·41-s − 0.218·43-s − 0.334·45-s − 1.51·47-s − 0.508·49-s + 0.924·51-s + 0.0445·53-s − 0.741·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1285632731\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1285632731\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 - T \) |
good | 3 | \( 1 + 0.869T + 3T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 + 5.49T + 11T^{2} \) |
| 13 | \( 1 + 0.851T + 13T^{2} \) |
| 17 | \( 1 + 7.59T + 17T^{2} \) |
| 19 | \( 1 + 6.07T + 19T^{2} \) |
| 23 | \( 1 + 3.96T + 23T^{2} \) |
| 29 | \( 1 - 2.47T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.01T + 37T^{2} \) |
| 41 | \( 1 + 6.31T + 41T^{2} \) |
| 43 | \( 1 + 1.43T + 43T^{2} \) |
| 47 | \( 1 + 10.4T + 47T^{2} \) |
| 53 | \( 1 - 0.324T + 53T^{2} \) |
| 59 | \( 1 + 3.54T + 59T^{2} \) |
| 61 | \( 1 - 6.61T + 61T^{2} \) |
| 67 | \( 1 + 10.3T + 67T^{2} \) |
| 71 | \( 1 - 1.96T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 1.12T + 79T^{2} \) |
| 83 | \( 1 - 8.15T + 83T^{2} \) |
| 89 | \( 1 - 9.14T + 89T^{2} \) |
| 97 | \( 1 - 0.182T + 97T^{2} \) |
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
−7.908814220993332732294013530069, −6.95721693166303448684917749990, −6.25304255317638824793257502670, −5.98300396696809496392592915993, −4.95941982017086619702939149903, −4.58928036154386323391540970309, −3.37978443815528256247493964279, −2.53558617791678817930119254310, −2.03076386905111932701073213926, −0.16496273833881025709493464290,
0.16496273833881025709493464290, 2.03076386905111932701073213926, 2.53558617791678817930119254310, 3.37978443815528256247493964279, 4.58928036154386323391540970309, 4.95941982017086619702939149903, 5.98300396696809496392592915993, 6.25304255317638824793257502670, 6.95721693166303448684917749990, 7.908814220993332732294013530069