| L(s) = 1 | + 1.61·2-s + 0.618·4-s − 2.23·5-s − 4.61·7-s − 2.23·8-s − 3.61·10-s + 2.23·11-s − 1.76·13-s − 7.47·14-s − 4.85·16-s + 4.85·17-s − 8.09·19-s − 1.38·20-s + 3.61·22-s + 2.38·23-s − 2.85·26-s − 2.85·28-s − 8.61·29-s + 9.56·31-s − 3.38·32-s + 7.85·34-s + 10.3·35-s − 6.85·37-s − 13.0·38-s + 5.00·40-s + 3.09·41-s + 4.70·43-s + ⋯ |
| L(s) = 1 | + 1.14·2-s + 0.309·4-s − 0.999·5-s − 1.74·7-s − 0.790·8-s − 1.14·10-s + 0.674·11-s − 0.489·13-s − 1.99·14-s − 1.21·16-s + 1.17·17-s − 1.85·19-s − 0.309·20-s + 0.771·22-s + 0.496·23-s − 0.559·26-s − 0.539·28-s − 1.60·29-s + 1.71·31-s − 0.597·32-s + 1.34·34-s + 1.74·35-s − 1.12·37-s − 2.12·38-s + 0.790·40-s + 0.482·41-s + 0.717·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 59 | \( 1 - T \) |
| good | 2 | \( 1 - 1.61T + 2T^{2} \) |
| 5 | \( 1 + 2.23T + 5T^{2} \) |
| 7 | \( 1 + 4.61T + 7T^{2} \) |
| 11 | \( 1 - 2.23T + 11T^{2} \) |
| 13 | \( 1 + 1.76T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 8.09T + 19T^{2} \) |
| 23 | \( 1 - 2.38T + 23T^{2} \) |
| 29 | \( 1 + 8.61T + 29T^{2} \) |
| 31 | \( 1 - 9.56T + 31T^{2} \) |
| 37 | \( 1 + 6.85T + 37T^{2} \) |
| 41 | \( 1 - 3.09T + 41T^{2} \) |
| 43 | \( 1 - 4.70T + 43T^{2} \) |
| 47 | \( 1 - 4.14T + 47T^{2} \) |
| 53 | \( 1 + 1.76T + 53T^{2} \) |
| 61 | \( 1 + 9.85T + 61T^{2} \) |
| 67 | \( 1 + 2.70T + 67T^{2} \) |
| 71 | \( 1 + 9.94T + 71T^{2} \) |
| 73 | \( 1 + 5.85T + 73T^{2} \) |
| 79 | \( 1 + 3T + 79T^{2} \) |
| 83 | \( 1 + 0.618T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 3T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.47530678490410855912690313326, −9.502296523815736970257957954829, −8.723522054716815213406115436304, −7.41637089409318599629907237811, −6.49163923820451365152842261989, −5.77374983578236085264445508964, −4.36348872594924038632514440715, −3.72053473065155089153147324641, −2.85185548347883289415635725978, 0,
2.85185548347883289415635725978, 3.72053473065155089153147324641, 4.36348872594924038632514440715, 5.77374983578236085264445508964, 6.49163923820451365152842261989, 7.41637089409318599629907237811, 8.723522054716815213406115436304, 9.502296523815736970257957954829, 10.47530678490410855912690313326
