| L(s) = 1 | − 2.08·2-s + 3·3-s − 3.64·4-s + 21.1·5-s − 6.26·6-s + 7·7-s + 24.3·8-s + 9·9-s − 44.0·10-s − 32.4·11-s − 10.9·12-s + 66.7·13-s − 14.6·14-s + 63.3·15-s − 21.5·16-s + 88.2·17-s − 18.7·18-s + 13.5·19-s − 77.0·20-s + 21·21-s + 67.6·22-s − 23·23-s + 72.9·24-s + 321.·25-s − 139.·26-s + 27·27-s − 25.5·28-s + ⋯ |
| L(s) = 1 | − 0.737·2-s + 0.577·3-s − 0.455·4-s + 1.88·5-s − 0.425·6-s + 0.377·7-s + 1.07·8-s + 0.333·9-s − 1.39·10-s − 0.888·11-s − 0.263·12-s + 1.42·13-s − 0.278·14-s + 1.09·15-s − 0.336·16-s + 1.25·17-s − 0.245·18-s + 0.163·19-s − 0.860·20-s + 0.218·21-s + 0.655·22-s − 0.208·23-s + 0.620·24-s + 2.57·25-s − 1.05·26-s + 0.192·27-s − 0.172·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.286054312\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.286054312\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 3T \) |
| 7 | \( 1 - 7T \) |
| 23 | \( 1 + 23T \) |
| good | 2 | \( 1 + 2.08T + 8T^{2} \) |
| 5 | \( 1 - 21.1T + 125T^{2} \) |
| 11 | \( 1 + 32.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 66.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 13.5T + 6.85e3T^{2} \) |
| 29 | \( 1 + 151.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 56.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 327.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 200.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 298.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 289.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 157.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 647.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 384.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 1.03e3T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.18e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.05e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 677.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 75.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−10.17194058002427930754610808069, −9.760092545746325767810867888988, −8.756676202184330263183820533051, −8.271081535274785868196125863739, −7.08370490084487816015352123064, −5.74425633012926563020867868369, −5.11682308328093438631697080365, −3.50187797524675567098220734607, −2.02705695468986189834885710293, −1.16259555132268138694202892610,
1.16259555132268138694202892610, 2.02705695468986189834885710293, 3.50187797524675567098220734607, 5.11682308328093438631697080365, 5.74425633012926563020867868369, 7.08370490084487816015352123064, 8.271081535274785868196125863739, 8.756676202184330263183820533051, 9.760092545746325767810867888988, 10.17194058002427930754610808069
