| L(s) = 1 | + 43.1·2-s + 171.·3-s + 1.34e3·4-s − 1.53e3·5-s + 7.37e3·6-s − 3.02e3·7-s + 3.60e4·8-s + 9.56e3·9-s − 6.62e4·10-s − 5.19e4·11-s + 2.30e5·12-s − 1.66e5·13-s − 1.30e5·14-s − 2.62e5·15-s + 8.63e5·16-s + 4.12e5·18-s − 1.03e6·19-s − 2.06e6·20-s − 5.17e5·21-s − 2.24e6·22-s + 6.47e5·23-s + 6.16e6·24-s + 4.06e5·25-s − 7.17e6·26-s − 1.73e6·27-s − 4.07e6·28-s − 1.01e5·29-s + ⋯ |
| L(s) = 1 | + 1.90·2-s + 1.21·3-s + 2.63·4-s − 1.09·5-s + 2.32·6-s − 0.476·7-s + 3.10·8-s + 0.486·9-s − 2.09·10-s − 1.07·11-s + 3.20·12-s − 1.61·13-s − 0.908·14-s − 1.33·15-s + 3.29·16-s + 0.926·18-s − 1.82·19-s − 2.89·20-s − 0.581·21-s − 2.03·22-s + 0.482·23-s + 3.79·24-s + 0.208·25-s − 3.07·26-s − 0.626·27-s − 1.25·28-s − 0.0265·29-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 | \( 1 - 43.1T + 512T^{2} \) |
| 3 | \( 1 - 171.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 1.53e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 3.02e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.19e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.66e5T + 1.06e10T^{2} \) |
| 19 | \( 1 + 1.03e6T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.47e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 1.01e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 1.03e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 5.58e6T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.18e6T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.60e6T + 5.02e14T^{2} \) |
| 47 | \( 1 - 3.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.22e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.60e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.86e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 3.73e7T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.04e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.95e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.96e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 3.93e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 8.75e8T + 7.60e17T^{2} \) |
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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
−9.994070379060058894739596344616, −8.415201388504999629857206796390, −7.57278310829527133255230571504, −6.87995150610994964358524544237, −5.48177794056703308362923212741, −4.43099954155689449575546663823, −3.72809280334629403190039954737, −2.68060578827006011066809709705, −2.29893781939393587451482495012, 0,
2.29893781939393587451482495012, 2.68060578827006011066809709705, 3.72809280334629403190039954737, 4.43099954155689449575546663823, 5.48177794056703308362923212741, 6.87995150610994964358524544237, 7.57278310829527133255230571504, 8.415201388504999629857206796390, 9.994070379060058894739596344616
