L(s) = 1 | + 6.98·2-s + 16.8·4-s − 77.5·5-s − 88.8·7-s − 106.·8-s − 541.·10-s − 322.·11-s − 425.·13-s − 620.·14-s − 1.27e3·16-s − 1.07e3·17-s + 1.04e3·19-s − 1.30e3·20-s − 2.25e3·22-s + 3.41e3·23-s + 2.88e3·25-s − 2.97e3·26-s − 1.49e3·28-s − 8.31e3·29-s − 9.59e3·31-s − 5.54e3·32-s − 7.51e3·34-s + 6.88e3·35-s − 1.01e3·37-s + 7.33e3·38-s + 8.22e3·40-s + 1.30e4·41-s + ⋯ |
L(s) = 1 | + 1.23·2-s + 0.525·4-s − 1.38·5-s − 0.684·7-s − 0.586·8-s − 1.71·10-s − 0.804·11-s − 0.697·13-s − 0.845·14-s − 1.24·16-s − 0.902·17-s + 0.666·19-s − 0.728·20-s − 0.993·22-s + 1.34·23-s + 0.922·25-s − 0.861·26-s − 0.359·28-s − 1.83·29-s − 1.79·31-s − 0.956·32-s − 1.11·34-s + 0.949·35-s − 0.121·37-s + 0.823·38-s + 0.812·40-s + 1.21·41-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
\(L(3)\) |
\(\approx\) |
\(0.8760784932\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8760784932\) |
\(L(\frac{7}{2})\) |
|
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 | \( 1 - 6.98T + 32T^{2} \) |
| 5 | \( 1 + 77.5T + 3.12e3T^{2} \) |
| 7 | \( 1 + 88.8T + 1.68e4T^{2} \) |
| 11 | \( 1 + 322.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 425.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.07e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.04e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.41e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 8.31e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.01e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 5.21e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.40e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.14e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.99e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 9.47e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 1.96e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.44e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 7.26e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.59e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.01e5T + 8.58e9T^{2} \) |
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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
−9.496847041323377611257088328706, −8.842539571170459470215730470209, −7.48511961895737944230057857079, −7.13377988443172732051464412415, −5.78624516881367461911455817881, −5.01368384808253870121074888254, −4.07395004631233286005134826353, −3.38948687807783015888318074273, −2.46969052509579298972497192797, −0.33668515716549539485906006728,
0.33668515716549539485906006728, 2.46969052509579298972497192797, 3.38948687807783015888318074273, 4.07395004631233286005134826353, 5.01368384808253870121074888254, 5.78624516881367461911455817881, 7.13377988443172732051464412415, 7.48511961895737944230057857079, 8.842539571170459470215730470209, 9.496847041323377611257088328706