| L(s) = 1 | − 6.91·2-s + 15.7·4-s + 11.1·5-s − 193.·7-s + 112.·8-s − 76.9·10-s + 125.·11-s + 1.11e3·13-s + 1.33e3·14-s − 1.27e3·16-s + 1.94e3·17-s − 2.11e3·19-s + 175.·20-s − 864.·22-s + 2.53e3·23-s − 3.00e3·25-s − 7.71e3·26-s − 3.05e3·28-s − 4.53e3·29-s − 902.·31-s + 5.26e3·32-s − 1.34e4·34-s − 2.15e3·35-s + 1.52e4·37-s + 1.46e4·38-s + 1.24e3·40-s + 8.45e3·41-s + ⋯ |
| L(s) = 1 | − 1.22·2-s + 0.493·4-s + 0.199·5-s − 1.49·7-s + 0.619·8-s − 0.243·10-s + 0.311·11-s + 1.83·13-s + 1.82·14-s − 1.24·16-s + 1.63·17-s − 1.34·19-s + 0.0982·20-s − 0.380·22-s + 1.00·23-s − 0.960·25-s − 2.23·26-s − 0.735·28-s − 1.00·29-s − 0.168·31-s + 0.908·32-s − 1.99·34-s − 0.296·35-s + 1.82·37-s + 1.64·38-s + 0.123·40-s + 0.785·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.8660356186\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8660356186\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 + 3.48e3T \) |
| good | 2 | \( 1 + 6.91T + 32T^{2} \) |
| 5 | \( 1 - 11.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 193.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 125.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 1.11e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.94e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.53e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 4.53e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 902.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.52e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 9.99e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.70e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.33e4T + 4.18e8T^{2} \) |
| 61 | \( 1 + 2.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 9.69e3T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.70e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.08e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.19e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.86e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 2.14e4T + 8.58e9T^{2} \) |
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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
−9.780763555099355769497971847537, −9.312471750162072105837875098262, −8.457838461651646889879626884631, −7.57781949593354140822482350744, −6.45433492585586157917580993272, −5.85474625689431989592723520049, −4.10766860416185526138252095793, −3.17120124446110519712596457477, −1.58779043694232860502206061041, −0.58823000628063767091137218821,
0.58823000628063767091137218821, 1.58779043694232860502206061041, 3.17120124446110519712596457477, 4.10766860416185526138252095793, 5.85474625689431989592723520049, 6.45433492585586157917580993272, 7.57781949593354140822482350744, 8.457838461651646889879626884631, 9.312471750162072105837875098262, 9.780763555099355769497971847537
