| L(s) = 1 | − 4.58·2-s − 10.9·4-s − 95.1·5-s − 70.9·7-s + 197.·8-s + 436.·10-s + 633.·11-s + 262.·13-s + 325.·14-s − 553.·16-s − 1.90e3·17-s + 1.30e3·19-s + 1.04e3·20-s − 2.90e3·22-s + 1.02e3·23-s + 5.93e3·25-s − 1.20e3·26-s + 776.·28-s + 3.27e3·29-s − 5.58e3·31-s − 3.76e3·32-s + 8.73e3·34-s + 6.75e3·35-s − 5.03e3·37-s − 5.98e3·38-s − 1.87e4·40-s − 1.45e3·41-s + ⋯ |
| L(s) = 1 | − 0.811·2-s − 0.342·4-s − 1.70·5-s − 0.547·7-s + 1.08·8-s + 1.38·10-s + 1.57·11-s + 0.431·13-s + 0.443·14-s − 0.540·16-s − 1.59·17-s + 0.829·19-s + 0.582·20-s − 1.27·22-s + 0.402·23-s + 1.89·25-s − 0.350·26-s + 0.187·28-s + 0.724·29-s − 1.04·31-s − 0.649·32-s + 1.29·34-s + 0.931·35-s − 0.605·37-s − 0.672·38-s − 1.85·40-s − 0.134·41-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
| \(L(3)\) |
\(\approx\) |
\(0.4705375959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4705375959\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 4.58T + 32T^{2} \) |
| 5 | \( 1 + 95.1T + 3.12e3T^{2} \) |
| 7 | \( 1 + 70.9T + 1.68e4T^{2} \) |
| 11 | \( 1 - 633.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 262.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.90e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.30e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.58e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 5.03e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.45e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.16e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 5.99e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.05e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.63e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.18e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.96e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.11e5T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.97e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 4.06e4T + 8.58e9T^{2} \) |
| 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
−9.238631472977387796056136191549, −8.914321990242003001312787662459, −8.074387282943057081198704709147, −7.15100471497686805700599041794, −6.54056113586120667949348144266, −4.82391432054949392323418561670, −4.03951236490250402335471367009, −3.34971660683593002283778916416, −1.44248402073149203203938107418, −0.39410457979645651392607988374,
0.39410457979645651392607988374, 1.44248402073149203203938107418, 3.34971660683593002283778916416, 4.03951236490250402335471367009, 4.82391432054949392323418561670, 6.54056113586120667949348144266, 7.15100471497686805700599041794, 8.074387282943057081198704709147, 8.914321990242003001312787662459, 9.238631472977387796056136191549
