| L(s) = 1 | − 4.14·2-s − 14.7·4-s + 97.3·5-s + 216.·7-s + 194.·8-s − 404.·10-s − 226.·11-s − 218.·13-s − 896.·14-s − 332.·16-s + 715.·17-s + 497.·19-s − 1.44e3·20-s + 937.·22-s + 905.·23-s + 6.35e3·25-s + 905.·26-s − 3.19e3·28-s − 2.69e3·29-s + 8.92e3·31-s − 4.83e3·32-s − 2.96e3·34-s + 2.10e4·35-s + 3.60e3·37-s − 2.06e3·38-s + 1.89e4·40-s + 8.72e3·41-s + ⋯ |
| L(s) = 1 | − 0.733·2-s − 0.462·4-s + 1.74·5-s + 1.66·7-s + 1.07·8-s − 1.27·10-s − 0.563·11-s − 0.358·13-s − 1.22·14-s − 0.324·16-s + 0.600·17-s + 0.316·19-s − 0.805·20-s + 0.413·22-s + 0.357·23-s + 2.03·25-s + 0.262·26-s − 0.770·28-s − 0.594·29-s + 1.66·31-s − 0.834·32-s − 0.440·34-s + 2.90·35-s + 0.432·37-s − 0.232·38-s + 1.86·40-s + 0.810·41-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
| \(L(3)\) |
\(\approx\) |
\(2.561219969\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.561219969\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 4.14T + 32T^{2} \) |
| 5 | \( 1 - 97.3T + 3.12e3T^{2} \) |
| 7 | \( 1 - 216.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 226.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 218.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 715.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 497.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 905.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.69e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 8.92e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.60e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 8.72e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.40e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.94e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.11e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.08e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 1.73e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.45e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.79e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 3.48e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 7.05e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.10e5T + 8.58e9T^{2} \) |
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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
−9.621738546852245498164787619604, −8.874260833657425344031763937699, −8.053963682383148174199716471940, −7.33327184695651298926045881731, −5.89999338787940335474436360991, −5.16112311472305026420673615069, −4.51421455259206387304061247886, −2.60815240809076451730124119597, −1.64597143963972252631040023899, −0.928420425319268089010464644325,
0.928420425319268089010464644325, 1.64597143963972252631040023899, 2.60815240809076451730124119597, 4.51421455259206387304061247886, 5.16112311472305026420673615069, 5.89999338787940335474436360991, 7.33327184695651298926045881731, 8.053963682383148174199716471940, 8.874260833657425344031763937699, 9.621738546852245498164787619604
