| L(s) = 1 | − 1.61·2-s + 3-s + 0.618·4-s + 1.38·5-s − 1.61·6-s + 7-s + 2.23·8-s + 9-s − 2.23·10-s + 11-s + 0.618·12-s − 1.61·13-s − 1.61·14-s + 1.38·15-s − 4.85·16-s + 3.47·17-s − 1.61·18-s − 0.236·19-s + 0.854·20-s + 21-s − 1.61·22-s + 23-s + 2.23·24-s − 3.09·25-s + 2.61·26-s + 27-s + 0.618·28-s + ⋯ |
| L(s) = 1 | − 1.14·2-s + 0.577·3-s + 0.309·4-s + 0.618·5-s − 0.660·6-s + 0.377·7-s + 0.790·8-s + 0.333·9-s − 0.707·10-s + 0.301·11-s + 0.178·12-s − 0.448·13-s − 0.432·14-s + 0.356·15-s − 1.21·16-s + 0.842·17-s − 0.381·18-s − 0.0541·19-s + 0.190·20-s + 0.218·21-s − 0.344·22-s + 0.208·23-s + 0.456·24-s − 0.618·25-s + 0.513·26-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
| \(L(1)\) |
\(\approx\) |
\(1.092389688\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.092389688\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.38T + 5T^{2} \) |
| 11 | \( 1 - T + 11T^{2} \) |
| 13 | \( 1 + 1.61T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 + 0.236T + 19T^{2} \) |
| 29 | \( 1 - 6.23T + 29T^{2} \) |
| 31 | \( 1 - 4.70T + 31T^{2} \) |
| 37 | \( 1 + 4.23T + 37T^{2} \) |
| 41 | \( 1 - 5.47T + 41T^{2} \) |
| 43 | \( 1 - 2.85T + 43T^{2} \) |
| 47 | \( 1 + 1.70T + 47T^{2} \) |
| 53 | \( 1 - 11.0T + 53T^{2} \) |
| 59 | \( 1 + 1.38T + 59T^{2} \) |
| 61 | \( 1 + 1.14T + 61T^{2} \) |
| 67 | \( 1 - 3.09T + 67T^{2} \) |
| 71 | \( 1 - 5.32T + 71T^{2} \) |
| 73 | \( 1 - 6.23T + 73T^{2} \) |
| 79 | \( 1 + 9.76T + 79T^{2} \) |
| 83 | \( 1 + 0.0557T + 83T^{2} \) |
| 89 | \( 1 + 6.56T + 89T^{2} \) |
| 97 | \( 1 + 6.70T + 97T^{2} \) |
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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
−10.51956163342141614770249780371, −9.925266876211551147261266414720, −9.234324300939714846843309286488, −8.383307199888269603550199521902, −7.69029462585363045915401481963, −6.69895685576002503227698034939, −5.32511920719835292083451807746, −4.14945186551961343812603338772, −2.51826894006238550268856117590, −1.24614586394501466118696681400,
1.24614586394501466118696681400, 2.51826894006238550268856117590, 4.14945186551961343812603338772, 5.32511920719835292083451807746, 6.69895685576002503227698034939, 7.69029462585363045915401481963, 8.383307199888269603550199521902, 9.234324300939714846843309286488, 9.925266876211551147261266414720, 10.51956163342141614770249780371
