| L(s) = 1 | − 1.61·2-s − 3-s + 0.618·4-s − 0.618·5-s + 1.61·6-s − 7-s + 2.23·8-s + 9-s + 1.00·10-s + 2.23·11-s − 0.618·12-s − 2.38·13-s + 1.61·14-s + 0.618·15-s − 4.85·16-s + 6.70·17-s − 1.61·18-s − 3.47·19-s − 0.381·20-s + 21-s − 3.61·22-s − 23-s − 2.23·24-s − 4.61·25-s + 3.85·26-s − 27-s − 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s − 0.577·3-s + 0.309·4-s − 0.276·5-s + 0.660·6-s − 0.377·7-s + 0.790·8-s + 0.333·9-s + 0.316·10-s + 0.674·11-s − 0.178·12-s − 0.660·13-s + 0.432·14-s + 0.159·15-s − 1.21·16-s + 1.62·17-s − 0.381·18-s − 0.796·19-s − 0.0854·20-s + 0.218·21-s − 0.771·22-s − 0.208·23-s − 0.456·24-s − 0.923·25-s + 0.755·26-s − 0.192·27-s − 0.116·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 + 1.61T + 2T^{2} \) |
| 5 | \( 1 + 0.618T + 5T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 2.38T + 13T^{2} \) |
| 17 | \( 1 - 6.70T + 17T^{2} \) |
| 19 | \( 1 + 3.47T + 19T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 6.70T + 31T^{2} \) |
| 37 | \( 1 + 11T + 37T^{2} \) |
| 41 | \( 1 + 1.47T + 41T^{2} \) |
| 43 | \( 1 + 1.61T + 43T^{2} \) |
| 47 | \( 1 + 7.23T + 47T^{2} \) |
| 53 | \( 1 + 13.0T + 53T^{2} \) |
| 59 | \( 1 - 9.38T + 59T^{2} \) |
| 61 | \( 1 + 4.85T + 61T^{2} \) |
| 67 | \( 1 + 5.09T + 67T^{2} \) |
| 71 | \( 1 - 4.38T + 71T^{2} \) |
| 73 | \( 1 + 12.7T + 73T^{2} \) |
| 79 | \( 1 + 9.47T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 97 | \( 1 - 10.4T + 97T^{2} \) |
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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
−10.14537550371901374733916161598, −9.863206365776709529947375400318, −8.820775860580404723590433718057, −7.85811733320914678684367073479, −7.12127925952419627726429323147, −6.02296917538655672157109057355, −4.81033452073727822321584879007, −3.60949947899386907940474602157, −1.63316623178825698113233874014, 0,
1.63316623178825698113233874014, 3.60949947899386907940474602157, 4.81033452073727822321584879007, 6.02296917538655672157109057355, 7.12127925952419627726429323147, 7.85811733320914678684367073479, 8.820775860580404723590433718057, 9.863206365776709529947375400318, 10.14537550371901374733916161598
