| L(s) = 1 | + 8·2-s − 91·3-s + 64·4-s − 728·6-s + 722·7-s + 512·8-s + 6.09e3·9-s − 1.33e3·11-s − 5.82e3·12-s − 1.10e4·13-s + 5.77e3·14-s + 4.09e3·16-s + 1.72e4·17-s + 4.87e4·18-s − 9.28e3·19-s − 6.57e4·21-s − 1.06e4·22-s − 2.29e4·23-s − 4.65e4·24-s − 8.81e4·26-s − 3.55e5·27-s + 4.62e4·28-s + 1.34e5·29-s − 2.87e5·31-s + 3.27e4·32-s + 1.21e5·33-s + 1.37e5·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1.94·3-s + 1/2·4-s − 1.37·6-s + 0.795·7-s + 0.353·8-s + 2.78·9-s − 0.301·11-s − 0.972·12-s − 1.39·13-s + 0.562·14-s + 1/4·16-s + 0.849·17-s + 1.97·18-s − 0.310·19-s − 1.54·21-s − 0.213·22-s − 0.393·23-s − 0.687·24-s − 0.983·26-s − 3.47·27-s + 0.397·28-s + 1.02·29-s − 1.73·31-s + 0.176·32-s + 0.586·33-s + 0.600·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 550 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{3} T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 91 T + p^{7} T^{2} \) |
| 7 | \( 1 - 722 T + p^{7} T^{2} \) |
| 13 | \( 1 + 11020 T + p^{7} T^{2} \) |
| 17 | \( 1 - 17210 T + p^{7} T^{2} \) |
| 19 | \( 1 + 9288 T + p^{7} T^{2} \) |
| 23 | \( 1 + 22971 T + p^{7} T^{2} \) |
| 29 | \( 1 - 134272 T + p^{7} T^{2} \) |
| 31 | \( 1 + 287765 T + p^{7} T^{2} \) |
| 37 | \( 1 - 316397 T + p^{7} T^{2} \) |
| 41 | \( 1 + 335968 T + p^{7} T^{2} \) |
| 43 | \( 1 - 858110 T + p^{7} T^{2} \) |
| 47 | \( 1 + 587680 T + p^{7} T^{2} \) |
| 53 | \( 1 - 244238 T + p^{7} T^{2} \) |
| 59 | \( 1 + 163287 T + p^{7} T^{2} \) |
| 61 | \( 1 - 37660 p T + p^{7} T^{2} \) |
| 67 | \( 1 - 3428283 T + p^{7} T^{2} \) |
| 71 | \( 1 - 1542953 T + p^{7} T^{2} \) |
| 73 | \( 1 + 2216316 T + p^{7} T^{2} \) |
| 79 | \( 1 - 1526014 T + p^{7} T^{2} \) |
| 83 | \( 1 + 1650370 T + p^{7} T^{2} \) |
| 89 | \( 1 - 5760847 T + p^{7} T^{2} \) |
| 97 | \( 1 - 5750759 T + p^{7} T^{2} \) |
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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
−9.686574794452388152463226373771, −7.85971602177703717231467096094, −7.17491585815831900922688569366, −6.23313901137218535909264641013, −5.30231498458645271375746041976, −4.93083863987974559133625873490, −3.96659429458227352370193862693, −2.21897956518606241730740638180, −1.10298617820915972211480482212, 0,
1.10298617820915972211480482212, 2.21897956518606241730740638180, 3.96659429458227352370193862693, 4.93083863987974559133625873490, 5.30231498458645271375746041976, 6.23313901137218535909264641013, 7.17491585815831900922688569366, 7.85971602177703717231467096094, 9.686574794452388152463226373771
