| L(s) = 1 | − 4.29·2-s − 29.9·3-s − 13.5·4-s + 25·5-s + 128.·6-s + 45.8·7-s + 195.·8-s + 655.·9-s − 107.·10-s − 504.·11-s + 405.·12-s + 513.·13-s − 196.·14-s − 749.·15-s − 408.·16-s + 289·17-s − 2.81e3·18-s − 2.08e3·19-s − 337.·20-s − 1.37e3·21-s + 2.16e3·22-s + 4.71e3·23-s − 5.86e3·24-s + 625·25-s − 2.20e3·26-s − 1.23e4·27-s − 619.·28-s + ⋯ |
| L(s) = 1 | − 0.759·2-s − 1.92·3-s − 0.422·4-s + 0.447·5-s + 1.46·6-s + 0.353·7-s + 1.08·8-s + 2.69·9-s − 0.339·10-s − 1.25·11-s + 0.812·12-s + 0.842·13-s − 0.268·14-s − 0.859·15-s − 0.399·16-s + 0.242·17-s − 2.04·18-s − 1.32·19-s − 0.188·20-s − 0.679·21-s + 0.955·22-s + 1.85·23-s − 2.07·24-s + 0.200·25-s − 0.639·26-s − 3.26·27-s − 0.149·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - 25T \) |
| 17 | \( 1 - 289T \) |
| good | 2 | \( 1 + 4.29T + 32T^{2} \) |
| 3 | \( 1 + 29.9T + 243T^{2} \) |
| 7 | \( 1 - 45.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 504.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 513.T + 3.71e5T^{2} \) |
| 19 | \( 1 + 2.08e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.09e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.15e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 4.51e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.37e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.75e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.47e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.00e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 496.T + 7.14e8T^{2} \) |
| 61 | \( 1 + 1.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.11e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.12e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.31e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.20e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 6.39e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.45e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.31e5T + 8.58e9T^{2} \) |
| 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
−12.80900399990217662590899496467, −11.14860245229861200859533544794, −10.69999036653285506745597004508, −9.705063777513555867577753821282, −8.180471995299211461629536238653, −6.77556547866310259788242806965, −5.48032831592342779979239860711, −4.60356687996272939632252837322, −1.31036660008991691344705720935, 0,
1.31036660008991691344705720935, 4.60356687996272939632252837322, 5.48032831592342779979239860711, 6.77556547866310259788242806965, 8.180471995299211461629536238653, 9.705063777513555867577753821282, 10.69999036653285506745597004508, 11.14860245229861200859533544794, 12.80900399990217662590899496467
