| L(s) = 1 | − 2-s − 0.0780·3-s + 4-s + 5-s + 0.0780·6-s − 1.81·7-s − 8-s − 2.99·9-s − 10-s − 0.519·11-s − 0.0780·12-s − 2.29·13-s + 1.81·14-s − 0.0780·15-s + 16-s − 5.26·17-s + 2.99·18-s + 5.89·19-s + 20-s + 0.141·21-s + 0.519·22-s − 7.08·23-s + 0.0780·24-s + 25-s + 2.29·26-s + 0.467·27-s − 1.81·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.0450·3-s + 0.5·4-s + 0.447·5-s + 0.0318·6-s − 0.686·7-s − 0.353·8-s − 0.997·9-s − 0.316·10-s − 0.156·11-s − 0.0225·12-s − 0.635·13-s + 0.485·14-s − 0.0201·15-s + 0.250·16-s − 1.27·17-s + 0.705·18-s + 1.35·19-s + 0.223·20-s + 0.0309·21-s + 0.110·22-s − 1.47·23-s + 0.0159·24-s + 0.200·25-s + 0.449·26-s + 0.0900·27-s − 0.343·28-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
| \(L(1)\) |
\(\approx\) |
\(0.8041914578\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8041914578\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 0.0780T + 3T^{2} \) |
| 7 | \( 1 + 1.81T + 7T^{2} \) |
| 11 | \( 1 + 0.519T + 11T^{2} \) |
| 13 | \( 1 + 2.29T + 13T^{2} \) |
| 17 | \( 1 + 5.26T + 17T^{2} \) |
| 19 | \( 1 - 5.89T + 19T^{2} \) |
| 23 | \( 1 + 7.08T + 23T^{2} \) |
| 29 | \( 1 - 2.68T + 29T^{2} \) |
| 31 | \( 1 - 4.07T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 + 5.30T + 43T^{2} \) |
| 47 | \( 1 + 4.66T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.71T + 59T^{2} \) |
| 61 | \( 1 - 7.57T + 61T^{2} \) |
| 67 | \( 1 - 3.59T + 67T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 - 1.94T + 83T^{2} \) |
| 89 | \( 1 - 13.8T + 89T^{2} \) |
| 97 | \( 1 + 9.90T + 97T^{2} \) |
| 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
−8.335580420173046329417707092474, −7.997267032954661596597153320452, −6.82874037821622891023658038829, −6.45872968849958527922830551191, −5.59806612929000365994278069490, −4.86656132375167765394708425153, −3.60441885476572732236885483657, −2.73358177580441227004742128470, −2.04749704456696621581483732003, −0.54106577049150719717688651410,
0.54106577049150719717688651410, 2.04749704456696621581483732003, 2.73358177580441227004742128470, 3.60441885476572732236885483657, 4.86656132375167765394708425153, 5.59806612929000365994278069490, 6.45872968849958527922830551191, 6.82874037821622891023658038829, 7.997267032954661596597153320452, 8.335580420173046329417707092474
