L(s) = 1 | + 0.625·2-s − 1.60·4-s + 0.920·5-s − 7-s − 2.25·8-s + 0.576·10-s − 0.186·11-s + 3.40·13-s − 0.625·14-s + 1.80·16-s − 6.65·17-s − 6.87·19-s − 1.48·20-s − 0.116·22-s + 4.83·23-s − 4.15·25-s + 2.13·26-s + 1.60·28-s + 5.29·29-s − 3.20·31-s + 5.64·32-s − 4.16·34-s − 0.920·35-s − 5.49·37-s − 4.30·38-s − 2.07·40-s + 0.995·41-s + ⋯ |
L(s) = 1 | + 0.442·2-s − 0.804·4-s + 0.411·5-s − 0.377·7-s − 0.798·8-s + 0.182·10-s − 0.0562·11-s + 0.945·13-s − 0.167·14-s + 0.451·16-s − 1.61·17-s − 1.57·19-s − 0.331·20-s − 0.0248·22-s + 1.00·23-s − 0.830·25-s + 0.418·26-s + 0.303·28-s + 0.982·29-s − 0.574·31-s + 0.997·32-s − 0.714·34-s − 0.155·35-s − 0.903·37-s − 0.697·38-s − 0.328·40-s + 0.155·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.467065582\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.467065582\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 127 | \( 1 + T \) |
good | 2 | \( 1 - 0.625T + 2T^{2} \) |
| 5 | \( 1 - 0.920T + 5T^{2} \) |
| 11 | \( 1 + 0.186T + 11T^{2} \) |
| 13 | \( 1 - 3.40T + 13T^{2} \) |
| 17 | \( 1 + 6.65T + 17T^{2} \) |
| 19 | \( 1 + 6.87T + 19T^{2} \) |
| 23 | \( 1 - 4.83T + 23T^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 3.20T + 31T^{2} \) |
| 37 | \( 1 + 5.49T + 37T^{2} \) |
| 41 | \( 1 - 0.995T + 41T^{2} \) |
| 43 | \( 1 + 5.38T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 + 2.21T + 53T^{2} \) |
| 59 | \( 1 + 0.266T + 59T^{2} \) |
| 61 | \( 1 - 12.0T + 61T^{2} \) |
| 67 | \( 1 + 4.28T + 67T^{2} \) |
| 71 | \( 1 - 3.89T + 71T^{2} \) |
| 73 | \( 1 - 3.02T + 73T^{2} \) |
| 79 | \( 1 - 12.4T + 79T^{2} \) |
| 83 | \( 1 - 5.49T + 83T^{2} \) |
| 89 | \( 1 - 0.466T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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
−8.022453352586001759189723951029, −6.76532183626557679455311226967, −6.49349828197139543445492594731, −5.69939058612045440206090970427, −5.01193324557288925376002594195, −4.20449141305471370068661450632, −3.75985852926731010076019464109, −2.74504809569135795893515052704, −1.89186351658459916938617374211, −0.54946716621996483347040882571,
0.54946716621996483347040882571, 1.89186351658459916938617374211, 2.74504809569135795893515052704, 3.75985852926731010076019464109, 4.20449141305471370068661450632, 5.01193324557288925376002594195, 5.69939058612045440206090970427, 6.49349828197139543445492594731, 6.76532183626557679455311226967, 8.022453352586001759189723951029