L(s) = 1 | + 2-s + 2.63·3-s + 4-s − 5-s + 2.63·6-s + 0.203·7-s + 8-s + 3.94·9-s − 10-s + 4.83·11-s + 2.63·12-s − 13-s + 0.203·14-s − 2.63·15-s + 16-s − 0.0683·17-s + 3.94·18-s + 3.00·19-s − 20-s + 0.535·21-s + 4.83·22-s + 4.98·23-s + 2.63·24-s + 25-s − 26-s + 2.48·27-s + 0.203·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s − 0.447·5-s + 1.07·6-s + 0.0767·7-s + 0.353·8-s + 1.31·9-s − 0.316·10-s + 1.45·11-s + 0.760·12-s − 0.277·13-s + 0.0542·14-s − 0.680·15-s + 0.250·16-s − 0.0165·17-s + 0.929·18-s + 0.689·19-s − 0.223·20-s + 0.116·21-s + 1.03·22-s + 1.03·23-s + 0.537·24-s + 0.200·25-s − 0.196·26-s + 0.479·27-s + 0.0383·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.537886014\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.537886014\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 + T \) |
good | 3 | \( 1 - 2.63T + 3T^{2} \) |
| 7 | \( 1 - 0.203T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 17 | \( 1 + 0.0683T + 17T^{2} \) |
| 19 | \( 1 - 3.00T + 19T^{2} \) |
| 23 | \( 1 - 4.98T + 23T^{2} \) |
| 29 | \( 1 + 4.67T + 29T^{2} \) |
| 37 | \( 1 - 6.73T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 0.517T + 43T^{2} \) |
| 47 | \( 1 - 2.49T + 47T^{2} \) |
| 53 | \( 1 - 7.40T + 53T^{2} \) |
| 59 | \( 1 + 9.25T + 59T^{2} \) |
| 61 | \( 1 - 8.56T + 61T^{2} \) |
| 67 | \( 1 + 5.12T + 67T^{2} \) |
| 71 | \( 1 - 1.20T + 71T^{2} \) |
| 73 | \( 1 + 2.64T + 73T^{2} \) |
| 79 | \( 1 - 7.88T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 - 13.1T + 89T^{2} \) |
| 97 | \( 1 + 1.83T + 97T^{2} \) |
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
−8.399566772174344469502697066030, −7.68180104754358004561513356708, −7.08472374231593728423983499519, −6.40479159208358306258841269555, −5.26223946966435149461906038226, −4.41984412128042109993725296034, −3.63069521052672956646084222740, −3.23448033319172831544523815564, −2.21788518626850889625733199442, −1.26656015597652495730999352302,
1.26656015597652495730999352302, 2.21788518626850889625733199442, 3.23448033319172831544523815564, 3.63069521052672956646084222740, 4.41984412128042109993725296034, 5.26223946966435149461906038226, 6.40479159208358306258841269555, 7.08472374231593728423983499519, 7.68180104754358004561513356708, 8.399566772174344469502697066030