| L(s) = 1 | + 6.28·2-s + 7.49·4-s − 48.8·5-s − 157.·7-s − 153.·8-s − 307.·10-s − 739.·11-s − 630.·13-s − 989.·14-s − 1.20e3·16-s − 369.·17-s − 522.·19-s − 366.·20-s − 4.64e3·22-s − 1.71e3·23-s − 737.·25-s − 3.95e3·26-s − 1.18e3·28-s + 7.00e3·29-s + 7.48e3·31-s − 2.66e3·32-s − 2.32e3·34-s + 7.69e3·35-s − 4.82e3·37-s − 3.28e3·38-s + 7.52e3·40-s − 2.80e3·41-s + ⋯ |
| L(s) = 1 | + 1.11·2-s + 0.234·4-s − 0.874·5-s − 1.21·7-s − 0.850·8-s − 0.971·10-s − 1.84·11-s − 1.03·13-s − 1.34·14-s − 1.17·16-s − 0.309·17-s − 0.332·19-s − 0.204·20-s − 2.04·22-s − 0.676·23-s − 0.236·25-s − 1.14·26-s − 0.284·28-s + 1.54·29-s + 1.39·31-s − 0.459·32-s − 0.344·34-s + 1.06·35-s − 0.579·37-s − 0.368·38-s + 0.743·40-s − 0.260·41-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
| \(L(3)\) |
\(\approx\) |
\(0.09707959552\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09707959552\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 6.28T + 32T^{2} \) |
| 5 | \( 1 + 48.8T + 3.12e3T^{2} \) |
| 7 | \( 1 + 157.T + 1.68e4T^{2} \) |
| 11 | \( 1 + 739.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 630.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 369.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 522.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 7.00e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 7.48e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.82e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.80e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.46e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.89e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 5.38e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.97e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.30e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.70e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.58e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 4.29e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.61e4T + 8.58e9T^{2} \) |
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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
−9.895296417238407945275558048118, −8.634477573607057982716126835800, −7.82547017744070040002463657568, −6.82229134638595278914753120543, −5.94052338448941682875027121661, −4.93042288065949650622308963300, −4.26482167068055831960524200782, −3.09172875256769165122293202974, −2.60513907231398955438326168252, −0.11176842035578823709182533550,
0.11176842035578823709182533550, 2.60513907231398955438326168252, 3.09172875256769165122293202974, 4.26482167068055831960524200782, 4.93042288065949650622308963300, 5.94052338448941682875027121661, 6.82229134638595278914753120543, 7.82547017744070040002463657568, 8.634477573607057982716126835800, 9.895296417238407945275558048118
