| L(s) = 1 | + 32.3·2-s + 533.·4-s − 2.07e3·5-s − 1.05e4·7-s + 692.·8-s − 6.69e4·10-s + 3.16e4·11-s + 4.76e4·13-s − 3.40e5·14-s − 2.50e5·16-s − 3.44e4·17-s + 7.63e5·19-s − 1.10e6·20-s + 1.02e6·22-s − 1.48e6·23-s + 2.33e6·25-s + 1.53e6·26-s − 5.61e6·28-s − 4.80e6·29-s − 7.82e6·31-s − 8.46e6·32-s − 1.11e6·34-s + 2.17e7·35-s + 9.43e6·37-s + 2.46e7·38-s − 1.43e6·40-s − 1.22e7·41-s + ⋯ |
| L(s) = 1 | + 1.42·2-s + 1.04·4-s − 1.48·5-s − 1.65·7-s + 0.0598·8-s − 2.11·10-s + 0.651·11-s + 0.462·13-s − 2.36·14-s − 0.956·16-s − 0.100·17-s + 1.34·19-s − 1.54·20-s + 0.930·22-s − 1.10·23-s + 1.19·25-s + 0.660·26-s − 1.72·28-s − 1.26·29-s − 1.52·31-s − 1.42·32-s − 0.142·34-s + 2.45·35-s + 0.827·37-s + 1.91·38-s − 0.0886·40-s − 0.679·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 32.3T + 512T^{2} \) |
| 5 | \( 1 + 2.07e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.05e4T + 4.03e7T^{2} \) |
| 11 | \( 1 - 3.16e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 4.76e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.44e4T + 1.18e11T^{2} \) |
| 19 | \( 1 - 7.63e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.48e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.80e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 9.43e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + 1.22e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 3.74e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.78e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 2.84e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 4.45e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.81e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.53e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.55e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.26e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.01e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.59e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.49e8T + 7.60e17T^{2} \) |
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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
−14.58263373059015470017301210418, −13.24599156214998428173060506962, −12.30398716258338207386282083466, −11.37923595423464343740378403665, −9.314943640556671546391269863093, −7.29368919273501520822689348860, −5.94139963711553859948138082766, −3.98961922030644275337677923316, −3.30916906172802683919540456871, 0,
3.30916906172802683919540456871, 3.98961922030644275337677923316, 5.94139963711553859948138082766, 7.29368919273501520822689348860, 9.314943640556671546391269863093, 11.37923595423464343740378403665, 12.30398716258338207386282083466, 13.24599156214998428173060506962, 14.58263373059015470017301210418
