| L(s) = 1 | + 9.34·2-s − 9·3-s + 55.3·4-s + 69.4·5-s − 84.1·6-s + 8.69·7-s + 218.·8-s + 81·9-s + 649.·10-s + 121·11-s − 498.·12-s − 970.·13-s + 81.2·14-s − 625.·15-s + 268.·16-s − 424.·17-s + 757.·18-s − 1.43e3·19-s + 3.84e3·20-s − 78.2·21-s + 1.13e3·22-s + 2.85e3·23-s − 1.96e3·24-s + 1.69e3·25-s − 9.07e3·26-s − 729·27-s + 481.·28-s + ⋯ |
| L(s) = 1 | + 1.65·2-s − 0.577·3-s + 1.72·4-s + 1.24·5-s − 0.953·6-s + 0.0670·7-s + 1.20·8-s + 0.333·9-s + 2.05·10-s + 0.301·11-s − 0.998·12-s − 1.59·13-s + 0.110·14-s − 0.717·15-s + 0.261·16-s − 0.356·17-s + 0.550·18-s − 0.909·19-s + 2.14·20-s − 0.0387·21-s + 0.498·22-s + 1.12·23-s − 0.695·24-s + 0.543·25-s − 2.63·26-s − 0.192·27-s + 0.115·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.427396878\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.427396878\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + 9T \) |
| 11 | \( 1 - 121T \) |
| good | 2 | \( 1 - 9.34T + 32T^{2} \) |
| 5 | \( 1 - 69.4T + 3.12e3T^{2} \) |
| 7 | \( 1 - 8.69T + 1.68e4T^{2} \) |
| 13 | \( 1 + 970.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 424.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.43e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.85e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.46e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 1.03e4T + 2.86e7T^{2} \) |
| 37 | \( 1 - 167.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 5.68e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.11e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 9.78e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.56e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.34e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.85e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.28e3T + 1.80e9T^{2} \) |
| 73 | \( 1 - 2.95e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 1.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.83e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.31e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 8.18e4T + 8.58e9T^{2} \) |
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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
−15.23431867127850081414487985713, −14.34243030527849428448463305240, −13.22545181873379446567013817213, −12.39609326600468949754693355121, −11.08882777826823977393015283963, −9.614806171845037458059377460363, −6.88320511046792337825395795697, −5.73734338741906870614759001660, −4.60762522429610294231293191605, −2.35251158114376928065764876110,
2.35251158114376928065764876110, 4.60762522429610294231293191605, 5.73734338741906870614759001660, 6.88320511046792337825395795697, 9.614806171845037458059377460363, 11.08882777826823977393015283963, 12.39609326600468949754693355121, 13.22545181873379446567013817213, 14.34243030527849428448463305240, 15.23431867127850081414487985713
