L(s) = 1 | + 2.73·2-s + 5.47·4-s − 4.25·5-s + 7-s + 9.49·8-s − 11.6·10-s − 0.824·11-s − 6.07·13-s + 2.73·14-s + 15.0·16-s + 0.158·17-s + 2.99·19-s − 23.2·20-s − 2.25·22-s + 7.25·23-s + 13.0·25-s − 16.6·26-s + 5.47·28-s + 8.45·29-s + 3.64·31-s + 22.0·32-s + 0.432·34-s − 4.25·35-s − 7.96·37-s + 8.19·38-s − 40.3·40-s + 2.33·41-s + ⋯ |
L(s) = 1 | + 1.93·2-s + 2.73·4-s − 1.90·5-s + 0.377·7-s + 3.35·8-s − 3.67·10-s − 0.248·11-s − 1.68·13-s + 0.730·14-s + 3.75·16-s + 0.0383·17-s + 0.687·19-s − 5.20·20-s − 0.480·22-s + 1.51·23-s + 2.61·25-s − 3.25·26-s + 1.03·28-s + 1.56·29-s + 0.654·31-s + 3.89·32-s + 0.0741·34-s − 0.719·35-s − 1.30·37-s + 1.32·38-s − 6.38·40-s + 0.364·41-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
\(L(1)\) |
\(\approx\) |
\(5.599608561\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.599608561\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.73T + 2T^{2} \) |
| 5 | \( 1 + 4.25T + 5T^{2} \) |
| 11 | \( 1 + 0.824T + 11T^{2} \) |
| 13 | \( 1 + 6.07T + 13T^{2} \) |
| 17 | \( 1 - 0.158T + 17T^{2} \) |
| 19 | \( 1 - 2.99T + 19T^{2} \) |
| 23 | \( 1 - 7.25T + 23T^{2} \) |
| 29 | \( 1 - 8.45T + 29T^{2} \) |
| 31 | \( 1 - 3.64T + 31T^{2} \) |
| 37 | \( 1 + 7.96T + 37T^{2} \) |
| 41 | \( 1 - 2.33T + 41T^{2} \) |
| 43 | \( 1 + 7.65T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 - 4.08T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 + 10.9T + 61T^{2} \) |
| 67 | \( 1 + 8.96T + 67T^{2} \) |
| 71 | \( 1 - 13.5T + 71T^{2} \) |
| 73 | \( 1 - 3.98T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 - 14.9T + 83T^{2} \) |
| 89 | \( 1 - 2.30T + 89T^{2} \) |
| 97 | \( 1 - 12.0T + 97T^{2} \) |
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
−7.61314440948378914847965548326, −7.00386457439173431703347222561, −6.56750232813947936203299411243, −5.23895022202927622665735039274, −4.84629680560865756653806070633, −4.53966589559920400597368997798, −3.50400826421841080504352337451, −3.10691142522177840844961790885, −2.31634709768154299594672480993, −0.869172400260465356343915609560,
0.869172400260465356343915609560, 2.31634709768154299594672480993, 3.10691142522177840844961790885, 3.50400826421841080504352337451, 4.53966589559920400597368997798, 4.84629680560865756653806070633, 5.23895022202927622665735039274, 6.56750232813947936203299411243, 7.00386457439173431703347222561, 7.61314440948378914847965548326