| L(s) = 1 | − 5.08·2-s − 6.19·4-s + 16.0·5-s − 124.·7-s + 194.·8-s − 81.2·10-s − 569.·11-s + 506.·13-s + 632.·14-s − 787.·16-s + 802.·17-s + 1.98e3·19-s − 99.1·20-s + 2.89e3·22-s − 2.62e3·23-s − 2.86e3·25-s − 2.57e3·26-s + 770.·28-s − 2.86e3·29-s − 1.06e3·31-s − 2.20e3·32-s − 4.07e3·34-s − 1.99e3·35-s − 1.00e4·37-s − 1.00e4·38-s + 3.10e3·40-s + 8.26e3·41-s + ⋯ |
| L(s) = 1 | − 0.898·2-s − 0.193·4-s + 0.286·5-s − 0.960·7-s + 1.07·8-s − 0.257·10-s − 1.41·11-s + 0.830·13-s + 0.862·14-s − 0.768·16-s + 0.673·17-s + 1.26·19-s − 0.0554·20-s + 1.27·22-s − 1.03·23-s − 0.918·25-s − 0.745·26-s + 0.185·28-s − 0.632·29-s − 0.198·31-s − 0.381·32-s − 0.604·34-s − 0.274·35-s − 1.20·37-s − 1.13·38-s + 0.306·40-s + 0.767·41-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
| \(L(3)\) |
\(\approx\) |
\(0.6137607298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6137607298\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 5.08T + 32T^{2} \) |
| 5 | \( 1 - 16.0T + 3.12e3T^{2} \) |
| 7 | \( 1 + 124.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 569.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 506.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 802.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.98e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.62e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 1.06e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.00e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.26e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.66e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.75e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.50e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 7.85e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.40e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.22e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 8.56e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.16e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.63e5T + 8.58e9T^{2} \) |
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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
−9.605175665494310006747383757077, −8.968933161934749497319084884213, −7.83019096149827544797761359110, −7.47504648936717689563972874742, −6.02202554758185372061031710930, −5.37450130394881583649074779491, −4.01666798199891628027338840243, −2.97524449893972240921055648525, −1.64098597156375931195789368144, −0.42442990954518726939937317593,
0.42442990954518726939937317593, 1.64098597156375931195789368144, 2.97524449893972240921055648525, 4.01666798199891628027338840243, 5.37450130394881583649074779491, 6.02202554758185372061031710930, 7.47504648936717689563972874742, 7.83019096149827544797761359110, 8.968933161934749497319084884213, 9.605175665494310006747383757077
