| L(s) = 1 | + 0.730·2-s + 3·3-s − 7.46·4-s − 2.77·5-s + 2.19·6-s − 36.3·7-s − 11.3·8-s + 9·9-s − 2.02·10-s + 11·11-s − 22.3·12-s − 24.2·13-s − 26.5·14-s − 8.31·15-s + 51.4·16-s + 66.6·17-s + 6.57·18-s + 153.·19-s + 20.6·20-s − 109.·21-s + 8.03·22-s + 197.·23-s − 33.9·24-s − 117.·25-s − 17.6·26-s + 27·27-s + 271.·28-s + ⋯ |
| L(s) = 1 | + 0.258·2-s + 0.577·3-s − 0.933·4-s − 0.247·5-s + 0.149·6-s − 1.96·7-s − 0.499·8-s + 0.333·9-s − 0.0640·10-s + 0.301·11-s − 0.538·12-s − 0.516·13-s − 0.507·14-s − 0.143·15-s + 0.804·16-s + 0.950·17-s + 0.0861·18-s + 1.85·19-s + 0.231·20-s − 1.13·21-s + 0.0779·22-s + 1.79·23-s − 0.288·24-s − 0.938·25-s − 0.133·26-s + 0.192·27-s + 1.83·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 11 | \( 1 - 11T \) |
| 61 | \( 1 - 61T \) |
| good | 2 | \( 1 - 0.730T + 8T^{2} \) |
| 5 | \( 1 + 2.77T + 125T^{2} \) |
| 7 | \( 1 + 36.3T + 343T^{2} \) |
| 13 | \( 1 + 24.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 66.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 153.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 197.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 225.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 37.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 124.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 422.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 18.7T + 7.95e4T^{2} \) |
| 47 | \( 1 - 452.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 569.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 430.T + 2.05e5T^{2} \) |
| 67 | \( 1 + 692.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 441.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 623.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 554.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.18e3T + 9.12e5T^{2} \) |
| 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
−8.705206388157741189546092670780, −7.36573235844025373025077383799, −7.13948685354627720438685261898, −5.79949348177819143772324381360, −5.28930709657813521758559190965, −3.99215741632912694818185996677, −3.38363221144982933137353489521, −2.88435410323406665144199674620, −1.06051005289461078509638427573, 0,
1.06051005289461078509638427573, 2.88435410323406665144199674620, 3.38363221144982933137353489521, 3.99215741632912694818185996677, 5.28930709657813521758559190965, 5.79949348177819143772324381360, 7.13948685354627720438685261898, 7.36573235844025373025077383799, 8.705206388157741189546092670780
