L(s) = 1 | − 4.44·2-s − 2.04·3-s + 11.7·4-s − 5·5-s + 9.06·6-s − 16.0·7-s − 16.5·8-s − 22.8·9-s + 22.2·10-s + 11·11-s − 23.9·12-s − 2.33·13-s + 71.4·14-s + 10.2·15-s − 20.2·16-s − 20.5·17-s + 101.·18-s − 19·19-s − 58.6·20-s + 32.8·21-s − 48.8·22-s + 134.·23-s + 33.7·24-s + 25·25-s + 10.3·26-s + 101.·27-s − 188.·28-s + ⋯ |
L(s) = 1 | − 1.57·2-s − 0.392·3-s + 1.46·4-s − 0.447·5-s + 0.616·6-s − 0.869·7-s − 0.731·8-s − 0.845·9-s + 0.702·10-s + 0.301·11-s − 0.575·12-s − 0.0497·13-s + 1.36·14-s + 0.175·15-s − 0.316·16-s − 0.292·17-s + 1.32·18-s − 0.229·19-s − 0.655·20-s + 0.341·21-s − 0.473·22-s + 1.22·23-s + 0.287·24-s + 0.200·25-s + 0.0780·26-s + 0.724·27-s − 1.27·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 + 5T \) |
| 11 | \( 1 - 11T \) |
| 19 | \( 1 + 19T \) |
good | 2 | \( 1 + 4.44T + 8T^{2} \) |
| 3 | \( 1 + 2.04T + 27T^{2} \) |
| 7 | \( 1 + 16.0T + 343T^{2} \) |
| 13 | \( 1 + 2.33T + 2.19e3T^{2} \) |
| 17 | \( 1 + 20.5T + 4.91e3T^{2} \) |
| 23 | \( 1 - 134.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 151.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 87.5T + 5.06e4T^{2} \) |
| 41 | \( 1 - 361.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 267.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 95.5T + 1.03e5T^{2} \) |
| 53 | \( 1 - 33.9T + 1.48e5T^{2} \) |
| 59 | \( 1 + 618.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 148.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 672.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 287.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 696.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 599.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 679.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.39e3T + 9.12e5T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.219054855495859240607520027761, −8.437048217435447788619483517680, −7.63460603140447367057519761700, −6.75917084029033515512534824544, −6.11565059041645599376258241571, −4.84112337714649127728404159173, −3.46651671365808804642587080585, −2.38289110429017147760418681760, −0.885421156364398064204302182306, 0,
0.885421156364398064204302182306, 2.38289110429017147760418681760, 3.46651671365808804642587080585, 4.84112337714649127728404159173, 6.11565059041645599376258241571, 6.75917084029033515512534824544, 7.63460603140447367057519761700, 8.437048217435447788619483517680, 9.219054855495859240607520027761