| L(s) = 1 | − 21.9·2-s + 352.·4-s + 397.·5-s + 802.·7-s − 4.93e3·8-s − 8.71e3·10-s − 4.39e3·11-s − 5.18e3·13-s − 1.75e4·14-s + 6.30e4·16-s − 2.15e4·17-s − 3.55e4·19-s + 1.40e5·20-s + 9.64e4·22-s + 5.53e3·23-s + 7.97e4·25-s + 1.13e5·26-s + 2.83e5·28-s + 2.45e4·29-s + 2.29e5·31-s − 7.50e5·32-s + 4.73e5·34-s + 3.18e5·35-s + 3.26e5·37-s + 7.80e5·38-s − 1.96e6·40-s − 8.70e4·41-s + ⋯ |
| L(s) = 1 | − 1.93·2-s + 2.75·4-s + 1.42·5-s + 0.883·7-s − 3.40·8-s − 2.75·10-s − 0.996·11-s − 0.654·13-s − 1.71·14-s + 3.84·16-s − 1.06·17-s − 1.19·19-s + 3.91·20-s + 1.93·22-s + 0.0947·23-s + 1.02·25-s + 1.26·26-s + 2.43·28-s + 0.186·29-s + 1.38·31-s − 4.04·32-s + 2.06·34-s + 1.25·35-s + 1.05·37-s + 2.30·38-s − 4.84·40-s − 0.197·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 59 | \( 1 - 2.05e5T \) |
| good | 2 | \( 1 + 21.9T + 128T^{2} \) |
| 5 | \( 1 - 397.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 802.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.39e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 5.18e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.15e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 3.55e4T + 8.93e8T^{2} \) |
| 23 | \( 1 - 5.53e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.45e4T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.29e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.26e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 8.70e4T + 1.94e11T^{2} \) |
| 43 | \( 1 - 3.44e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 4.82e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.52e6T + 1.17e12T^{2} \) |
| 61 | \( 1 - 1.94e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 1.35e5T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.65e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.46e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.95e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 5.37e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 6.02e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.52e7T + 8.07e13T^{2} \) |
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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
−9.318594656925978563531701328213, −8.455811226760288037550860866215, −7.82143038133130439690600188276, −6.72136003683623413712555531922, −6.01788741929122247396455871500, −4.84714623170475621645876799098, −2.46535490872031758228919824506, −2.25987828306083638026054352639, −1.16307564214366347814922838519, 0,
1.16307564214366347814922838519, 2.25987828306083638026054352639, 2.46535490872031758228919824506, 4.84714623170475621645876799098, 6.01788741929122247396455871500, 6.72136003683623413712555531922, 7.82143038133130439690600188276, 8.455811226760288037550860866215, 9.318594656925978563531701328213
