| L(s) = 1 | + 2-s − 0.902·3-s + 4-s + 0.557·5-s − 0.902·6-s + 7-s + 8-s − 2.18·9-s + 0.557·10-s − 1.35·11-s − 0.902·12-s + 1.45·13-s + 14-s − 0.502·15-s + 16-s + 2.34·17-s − 2.18·18-s + 0.557·20-s − 0.902·21-s − 1.35·22-s + 3.35·23-s − 0.902·24-s − 4.68·25-s + 1.45·26-s + 4.67·27-s + 28-s − 0.804·29-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.520·3-s + 0.5·4-s + 0.249·5-s − 0.368·6-s + 0.377·7-s + 0.353·8-s − 0.728·9-s + 0.176·10-s − 0.407·11-s − 0.260·12-s + 0.404·13-s + 0.267·14-s − 0.129·15-s + 0.250·16-s + 0.568·17-s − 0.515·18-s + 0.124·20-s − 0.196·21-s − 0.288·22-s + 0.698·23-s − 0.184·24-s − 0.937·25-s + 0.286·26-s + 0.900·27-s + 0.188·28-s − 0.149·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.655643917\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.655643917\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 0.902T + 3T^{2} \) |
| 5 | \( 1 - 0.557T + 5T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 - 1.45T + 13T^{2} \) |
| 17 | \( 1 - 2.34T + 17T^{2} \) |
| 23 | \( 1 - 3.35T + 23T^{2} \) |
| 29 | \( 1 + 0.804T + 29T^{2} \) |
| 31 | \( 1 - 1.09T + 31T^{2} \) |
| 37 | \( 1 - 6.78T + 37T^{2} \) |
| 41 | \( 1 + 5.03T + 41T^{2} \) |
| 43 | \( 1 - 2.49T + 43T^{2} \) |
| 47 | \( 1 + 2.24T + 47T^{2} \) |
| 53 | \( 1 - 5.37T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 7.15T + 61T^{2} \) |
| 67 | \( 1 + 10.0T + 67T^{2} \) |
| 71 | \( 1 - 3.00T + 71T^{2} \) |
| 73 | \( 1 - 7.48T + 73T^{2} \) |
| 79 | \( 1 - 1.96T + 79T^{2} \) |
| 83 | \( 1 + 9.06T + 83T^{2} \) |
| 89 | \( 1 - 3.12T + 89T^{2} \) |
| 97 | \( 1 - 12.7T + 97T^{2} \) |
| 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
−8.152706232655742877414270649217, −7.43054229741332796857856992184, −6.58133879839270150327658720812, −5.85132130732452509057573921118, −5.41608685167298953714819761136, −4.71093634930739973230144983544, −3.77156651045790969730676136652, −2.92528775495107507666753693947, −2.05185018856517626124244202675, −0.822238766108433118248499429621,
0.822238766108433118248499429621, 2.05185018856517626124244202675, 2.92528775495107507666753693947, 3.77156651045790969730676136652, 4.71093634930739973230144983544, 5.41608685167298953714819761136, 5.85132130732452509057573921118, 6.58133879839270150327658720812, 7.43054229741332796857856992184, 8.152706232655742877414270649217
