| L(s) = 1 | − 8.08·2-s + 33.2·4-s − 19.8·5-s + 75.2·7-s − 10.4·8-s + 160.·10-s − 188.·11-s − 837.·13-s − 607.·14-s − 981.·16-s − 930.·17-s − 2.15e3·19-s − 661.·20-s + 1.52e3·22-s + 2.36e3·23-s − 2.72e3·25-s + 6.76e3·26-s + 2.50e3·28-s + 101.·29-s − 5.82e3·31-s + 8.26e3·32-s + 7.51e3·34-s − 1.49e3·35-s + 1.07e4·37-s + 1.74e4·38-s + 206.·40-s − 1.43e4·41-s + ⋯ |
| L(s) = 1 | − 1.42·2-s + 1.04·4-s − 0.355·5-s + 0.580·7-s − 0.0574·8-s + 0.507·10-s − 0.470·11-s − 1.37·13-s − 0.828·14-s − 0.958·16-s − 0.780·17-s − 1.37·19-s − 0.369·20-s + 0.671·22-s + 0.932·23-s − 0.873·25-s + 1.96·26-s + 0.603·28-s + 0.0224·29-s − 1.08·31-s + 1.42·32-s + 1.11·34-s − 0.206·35-s + 1.29·37-s + 1.95·38-s + 0.0204·40-s − 1.33·41-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
| \(L(3)\) |
\(\approx\) |
\(0.3026136596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3026136596\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 8.08T + 32T^{2} \) |
| 5 | \( 1 + 19.8T + 3.12e3T^{2} \) |
| 7 | \( 1 - 75.2T + 1.68e4T^{2} \) |
| 11 | \( 1 + 188.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 837.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 930.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 101.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.82e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.07e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.43e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 669.T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.84e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.53e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.53e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.96e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 6.12e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 5.10e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.53e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.50e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 9.55e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.36e4T + 8.58e9T^{2} \) |
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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
−9.464487449240828901760344446216, −8.843381948233619068357989771424, −7.919970176687126842556039343785, −7.45442017055568498280559029671, −6.49567079651670554034675279192, −5.05261548596759214353551680821, −4.25408781380575844786903132534, −2.55626364816610199163392305984, −1.73322194179309740187532922813, −0.31024839192029521408980662928,
0.31024839192029521408980662928, 1.73322194179309740187532922813, 2.55626364816610199163392305984, 4.25408781380575844786903132534, 5.05261548596759214353551680821, 6.49567079651670554034675279192, 7.45442017055568498280559029671, 7.919970176687126842556039343785, 8.843381948233619068357989771424, 9.464487449240828901760344446216
