| L(s) = 1 | − 2·2-s − 3·3-s + 4·4-s + 10.3·5-s + 6·6-s + 7·7-s − 8·8-s + 9·9-s − 20.6·10-s − 26.5·11-s − 12·12-s + 51.6·13-s − 14·14-s − 30.9·15-s + 16·16-s − 11.0·17-s − 18·18-s − 54.2·19-s + 41.2·20-s − 21·21-s + 53.1·22-s − 23·23-s + 24·24-s − 18.5·25-s − 103.·26-s − 27·27-s + 28·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.922·5-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 0.333·9-s − 0.652·10-s − 0.728·11-s − 0.288·12-s + 1.10·13-s − 0.267·14-s − 0.532·15-s + 0.250·16-s − 0.157·17-s − 0.235·18-s − 0.655·19-s + 0.461·20-s − 0.218·21-s + 0.515·22-s − 0.208·23-s + 0.204·24-s − 0.148·25-s − 0.778·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 5 | \( 1 - 10.3T + 125T^{2} \) |
| 11 | \( 1 + 26.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 51.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 11.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 54.2T + 6.85e3T^{2} \) |
| 29 | \( 1 + 310.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 146.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 230.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 102.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 224.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 197.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 577.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 360.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 24.5T + 2.26e5T^{2} \) |
| 67 | \( 1 - 69.1T + 3.00e5T^{2} \) |
| 71 | \( 1 - 213.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 308.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 134.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 165.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 431.T + 9.12e5T^{2} \) |
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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
−9.299911242361300121297776376603, −8.441384820884540516328954176598, −7.63346091422127589181179585898, −6.55505929006815633271945961806, −5.88392395131340676189445257791, −5.10695312098694980786235740444, −3.75796591988116668203016820672, −2.26937679631435045335834660510, −1.41961445519364428088706393944, 0,
1.41961445519364428088706393944, 2.26937679631435045335834660510, 3.75796591988116668203016820672, 5.10695312098694980786235740444, 5.88392395131340676189445257791, 6.55505929006815633271945961806, 7.63346091422127589181179585898, 8.441384820884540516328954176598, 9.299911242361300121297776376603
