| L(s) = 1 | + 2-s + 4-s − 3·7-s + 8-s − 2·9-s − 3·14-s + 16-s − 11·17-s − 2·18-s + 11·23-s + 25-s − 3·28-s + 6·31-s + 32-s − 11·34-s − 2·36-s − 17·41-s + 11·46-s − 16·47-s + 2·49-s + 50-s − 3·56-s + 6·62-s + 6·63-s + 64-s − 11·68-s − 13·71-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 1.13·7-s + 0.353·8-s − 2/3·9-s − 0.801·14-s + 1/4·16-s − 2.66·17-s − 0.471·18-s + 2.29·23-s + 1/5·25-s − 0.566·28-s + 1.07·31-s + 0.176·32-s − 1.88·34-s − 1/3·36-s − 2.65·41-s + 1.62·46-s − 2.33·47-s + 2/7·49-s + 0.141·50-s − 0.400·56-s + 0.762·62-s + 0.755·63-s + 1/8·64-s − 1.33·68-s − 1.54·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 3 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 15 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 56 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 80 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 67 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.927813633752179218126680210825, −8.627435141432831789673869029474, −8.210628309357274172716467685518, −7.24937595111234950787191519988, −6.85088256909581415242634775790, −6.53188993190622290430180289888, −6.24556395238742447064275332072, −5.38795437512056532874555005533, −4.82214958135051684697679126838, −4.54938958466510561165188561616, −3.64011646744725010470789894465, −2.99306507198769018152842120753, −2.72987143554639332798675974849, −1.66855124516115900754616347362, 0,
1.66855124516115900754616347362, 2.72987143554639332798675974849, 2.99306507198769018152842120753, 3.64011646744725010470789894465, 4.54938958466510561165188561616, 4.82214958135051684697679126838, 5.38795437512056532874555005533, 6.24556395238742447064275332072, 6.53188993190622290430180289888, 6.85088256909581415242634775790, 7.24937595111234950787191519988, 8.210628309357274172716467685518, 8.627435141432831789673869029474, 8.927813633752179218126680210825
