| L(s) = 1 | − 2·2-s + 2·4-s − 2·7-s − 5·9-s + 4·14-s − 4·16-s + 11·17-s + 10·18-s − 7·23-s + 7·25-s − 4·28-s − 18·31-s + 8·32-s − 22·34-s − 10·36-s + 11·41-s + 14·46-s − 4·47-s − 7·49-s − 14·50-s + 36·62-s + 10·63-s − 8·64-s + 22·68-s + 6·71-s − 13·73-s − 9·79-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 4-s − 0.755·7-s − 5/3·9-s + 1.06·14-s − 16-s + 2.66·17-s + 2.35·18-s − 1.45·23-s + 7/5·25-s − 0.755·28-s − 3.23·31-s + 1.41·32-s − 3.77·34-s − 5/3·36-s + 1.71·41-s + 2.06·46-s − 0.583·47-s − 49-s − 1.97·50-s + 4.57·62-s + 1.25·63-s − 64-s + 2.66·68-s + 0.712·71-s − 1.52·73-s − 1.01·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46016 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46016 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 719 | $C_1$$\times$$C_2$ | \( ( 1 - T )( 1 + 18 T + p T^{2} ) \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 27 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.675608253667301734993955346566, −9.507749555405025129686767504313, −8.911258647527967813501217674872, −8.425479657364013638685351439996, −7.906643032230706506285526016482, −7.54183264572149741276619273780, −6.94438396048493634466287737983, −6.15790300508413636406777969666, −5.60452706764639161595334207501, −5.30738068330187321234687195572, −4.02646008028666957236840566486, −3.28438111144352555354874763920, −2.69702239590077889294438943456, −1.46138740786072346597089115812, 0,
1.46138740786072346597089115812, 2.69702239590077889294438943456, 3.28438111144352555354874763920, 4.02646008028666957236840566486, 5.30738068330187321234687195572, 5.60452706764639161595334207501, 6.15790300508413636406777969666, 6.94438396048493634466287737983, 7.54183264572149741276619273780, 7.906643032230706506285526016482, 8.425479657364013638685351439996, 8.911258647527967813501217674872, 9.507749555405025129686767504313, 9.675608253667301734993955346566
