| L(s) = 1 | − 9.35·2-s + 55.5·4-s + 63.9·5-s + 207.·7-s − 220.·8-s − 598.·10-s + 221.·11-s − 10.6·13-s − 1.93e3·14-s + 287.·16-s + 1.73e3·17-s − 2.11e3·19-s + 3.55e3·20-s − 2.07e3·22-s − 2.69e3·23-s + 959.·25-s + 100.·26-s + 1.15e4·28-s − 2.67e3·29-s − 6.09e3·31-s + 4.37e3·32-s − 1.62e4·34-s + 1.32e4·35-s + 5.66e3·37-s + 1.97e4·38-s − 1.41e4·40-s + 8.02e3·41-s + ⋯ |
| L(s) = 1 | − 1.65·2-s + 1.73·4-s + 1.14·5-s + 1.59·7-s − 1.22·8-s − 1.89·10-s + 0.551·11-s − 0.0175·13-s − 2.64·14-s + 0.281·16-s + 1.45·17-s − 1.34·19-s + 1.98·20-s − 0.912·22-s − 1.06·23-s + 0.307·25-s + 0.0290·26-s + 2.77·28-s − 0.589·29-s − 1.13·31-s + 0.754·32-s − 2.41·34-s + 1.82·35-s + 0.680·37-s + 2.21·38-s − 1.39·40-s + 0.745·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.676441864\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.676441864\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 9.35T + 32T^{2} \) |
| 5 | \( 1 - 63.9T + 3.12e3T^{2} \) |
| 7 | \( 1 - 207.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 221.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 10.6T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.73e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.69e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.67e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 6.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.66e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.02e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.10e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.21e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.86e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.65e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.91e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.76e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.92e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.35e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.28e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.69e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 3.48e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.76e3T + 8.58e9T^{2} \) |
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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
−9.517815234445161388487989675499, −8.889275372700844385023487570664, −7.999892390811646236857243145484, −7.47790417840600066290528067590, −6.23436197020541093160367183392, −5.47383074298732466679372009308, −4.13562075546686682506134653176, −2.22407494918787073519019477577, −1.74387670174608726266780138761, −0.816899656250716559898207228005,
0.816899656250716559898207228005, 1.74387670174608726266780138761, 2.22407494918787073519019477577, 4.13562075546686682506134653176, 5.47383074298732466679372009308, 6.23436197020541093160367183392, 7.47790417840600066290528067590, 7.999892390811646236857243145484, 8.889275372700844385023487570664, 9.517815234445161388487989675499
