| L(s) = 1 | − 22.4·2-s − 8·4-s + 493.·5-s − 763·7-s + 1.16e4·8-s − 1.10e4·10-s + 5.68e4·11-s − 7.30e4·13-s + 1.71e4·14-s − 2.57e5·16-s − 1.68e5·17-s − 5.98e5·19-s − 3.95e3·20-s − 1.27e6·22-s − 2.40e6·23-s − 1.70e6·25-s + 1.63e6·26-s + 6.10e3·28-s + 4.65e6·29-s − 1.82e6·31-s − 1.85e5·32-s + 3.77e6·34-s − 3.76e5·35-s + 1.42e7·37-s + 1.34e7·38-s + 5.76e6·40-s − 2.98e7·41-s + ⋯ |
| L(s) = 1 | − 0.992·2-s − 0.0156·4-s + 0.353·5-s − 0.120·7-s + 1.00·8-s − 0.350·10-s + 1.17·11-s − 0.709·13-s + 0.119·14-s − 0.984·16-s − 0.488·17-s − 1.05·19-s − 0.00552·20-s − 1.16·22-s − 1.78·23-s − 0.875·25-s + 0.703·26-s + 0.00187·28-s + 1.22·29-s − 0.354·31-s − 0.0312·32-s + 0.484·34-s − 0.0424·35-s + 1.25·37-s + 1.04·38-s + 0.356·40-s − 1.64·41-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 | \( 1 + 22.4T + 512T^{2} \) |
| 5 | \( 1 - 493.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 763T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.68e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 7.30e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 1.68e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 5.98e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.40e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.65e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.82e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.42e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + 2.98e7T + 3.27e14T^{2} \) |
| 43 | \( 1 - 7.75e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.09e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.90e6T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.25e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.52e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.20e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 1.09e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 6.64e7T + 5.88e16T^{2} \) |
| 79 | \( 1 + 1.15e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.14e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 3.37e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 2.35e8T + 7.60e17T^{2} \) |
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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
−14.61360759917836670001478036997, −13.46107226578161409422280523067, −11.88887391690662373813774350240, −10.28215311961379465954612540009, −9.342899165113241410791257358408, −8.098890245276747473166139813864, −6.45504727174199469632692837404, −4.33292616083138530350317055849, −1.80625810749324664693601615510, 0,
1.80625810749324664693601615510, 4.33292616083138530350317055849, 6.45504727174199469632692837404, 8.098890245276747473166139813864, 9.342899165113241410791257358408, 10.28215311961379465954612540009, 11.88887391690662373813774350240, 13.46107226578161409422280523067, 14.61360759917836670001478036997
