| L(s) = 1 | − 7.00·2-s − 248.·3-s − 462.·4-s + 1.62e3·5-s + 1.74e3·6-s + 8.78e3·7-s + 6.83e3·8-s + 4.21e4·9-s − 1.13e4·10-s + 5.24e4·11-s + 1.15e5·12-s + 1.44e4·13-s − 6.15e4·14-s − 4.03e5·15-s + 1.89e5·16-s − 2.95e5·18-s − 9.31e5·19-s − 7.51e5·20-s − 2.18e6·21-s − 3.67e5·22-s − 1.37e6·23-s − 1.69e6·24-s + 6.85e5·25-s − 1.01e5·26-s − 5.57e6·27-s − 4.06e6·28-s + 4.42e6·29-s + ⋯ |
| L(s) = 1 | − 0.309·2-s − 1.77·3-s − 0.904·4-s + 1.16·5-s + 0.548·6-s + 1.38·7-s + 0.589·8-s + 2.13·9-s − 0.360·10-s + 1.08·11-s + 1.60·12-s + 0.140·13-s − 0.428·14-s − 2.05·15-s + 0.721·16-s − 0.662·18-s − 1.64·19-s − 1.05·20-s − 2.44·21-s − 0.334·22-s − 1.02·23-s − 1.04·24-s + 0.351·25-s − 0.0434·26-s − 2.01·27-s − 1.24·28-s + 1.16·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 289 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 17 | \( 1 \) |
| good | 2 | \( 1 + 7.00T + 512T^{2} \) |
| 3 | \( 1 + 248.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.62e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 8.78e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.24e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.44e4T + 1.06e10T^{2} \) |
| 19 | \( 1 + 9.31e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.37e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.42e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.02e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 1.55e7T + 1.29e14T^{2} \) |
| 41 | \( 1 - 8.35e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.42e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.77e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.77e6T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.56e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.14e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 5.42e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.38e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 3.24e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.05e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 1.35e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.08e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + 7.22e8T + 7.60e17T^{2} \) |
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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
−10.01060322491729226551613693791, −8.960058932219276956716573607507, −7.898903783064640044336312686016, −6.38034368727241806343280544039, −5.91515456200709808891158152678, −4.70692100751197619843778513363, −4.38177946862741585287223279677, −1.74765619843517159730308181683, −1.19174248585652055877549941201, 0,
1.19174248585652055877549941201, 1.74765619843517159730308181683, 4.38177946862741585287223279677, 4.70692100751197619843778513363, 5.91515456200709808891158152678, 6.38034368727241806343280544039, 7.898903783064640044336312686016, 8.960058932219276956716573607507, 10.01060322491729226551613693791
