| L(s) = 1 | + 2-s + 0.642·3-s + 4-s − 5-s + 0.642·6-s + 4.88·7-s + 8-s − 2.58·9-s − 10-s − 4.59·11-s + 0.642·12-s + 0.824·13-s + 4.88·14-s − 0.642·15-s + 16-s + 5.12·17-s − 2.58·18-s − 2.57·19-s − 20-s + 3.13·21-s − 4.59·22-s + 3.63·23-s + 0.642·24-s + 25-s + 0.824·26-s − 3.58·27-s + 4.88·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.370·3-s + 0.5·4-s − 0.447·5-s + 0.262·6-s + 1.84·7-s + 0.353·8-s − 0.862·9-s − 0.316·10-s − 1.38·11-s + 0.185·12-s + 0.228·13-s + 1.30·14-s − 0.165·15-s + 0.250·16-s + 1.24·17-s − 0.609·18-s − 0.589·19-s − 0.223·20-s + 0.684·21-s − 0.980·22-s + 0.758·23-s + 0.131·24-s + 0.200·25-s + 0.161·26-s − 0.690·27-s + 0.922·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.713418473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.713418473\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 3 | \( 1 - 0.642T + 3T^{2} \) |
| 7 | \( 1 - 4.88T + 7T^{2} \) |
| 11 | \( 1 + 4.59T + 11T^{2} \) |
| 13 | \( 1 - 0.824T + 13T^{2} \) |
| 17 | \( 1 - 5.12T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 - 3.63T + 23T^{2} \) |
| 29 | \( 1 - 4.06T + 29T^{2} \) |
| 31 | \( 1 - 8.34T + 31T^{2} \) |
| 37 | \( 1 + 6.23T + 37T^{2} \) |
| 41 | \( 1 + 0.0307T + 41T^{2} \) |
| 43 | \( 1 - 2.80T + 43T^{2} \) |
| 47 | \( 1 - 6.02T + 47T^{2} \) |
| 53 | \( 1 - 10.9T + 53T^{2} \) |
| 59 | \( 1 + 1.76T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 + 3.48T + 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 1.60T + 79T^{2} \) |
| 83 | \( 1 - 9.48T + 83T^{2} \) |
| 89 | \( 1 - 9.44T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
| 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
−8.374527061582507295001325977489, −7.77695089263797325694414111136, −7.20135804732549457004511855856, −5.95744802910422720495783711137, −5.25463743783742255486086476883, −4.82891364015731655233301874284, −3.90765744590822613321518998536, −2.89569718502859949504812458499, −2.28542157314538655802897221406, −1.01516179911229844522157238476,
1.01516179911229844522157238476, 2.28542157314538655802897221406, 2.89569718502859949504812458499, 3.90765744590822613321518998536, 4.82891364015731655233301874284, 5.25463743783742255486086476883, 5.95744802910422720495783711137, 7.20135804732549457004511855856, 7.77695089263797325694414111136, 8.374527061582507295001325977489
