L(s) = 1 | − 2-s − 4-s + 2·7-s + 8-s − 2·9-s + 3·11-s − 2·14-s + 3·16-s + 2·18-s − 3·22-s + 6·23-s + 2·25-s − 2·28-s + 3·29-s − 3·32-s + 2·36-s + 13·37-s − 11·43-s − 3·44-s − 6·46-s − 3·49-s − 2·50-s − 3·53-s + 2·56-s − 3·58-s − 4·63-s − 5·64-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.755·7-s + 0.353·8-s − 2/3·9-s + 0.904·11-s − 0.534·14-s + 3/4·16-s + 0.471·18-s − 0.639·22-s + 1.25·23-s + 2/5·25-s − 0.377·28-s + 0.557·29-s − 0.530·32-s + 1/3·36-s + 2.13·37-s − 1.67·43-s − 0.452·44-s − 0.884·46-s − 3/7·49-s − 0.282·50-s − 0.412·53-s + 0.267·56-s − 0.393·58-s − 0.503·63-s − 5/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11858 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6681225335\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6681225335\) |
\(L(\frac{3}{2})\) |
|
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$$\times$$C_2$ | \( ( 1 + T )( 1 + p T^{2} ) \) |
| 7 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 11 | $C_2$ | \( 1 - 3 T + p T^{2} \) |
good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 41 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 53 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 68 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 64 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 91 T^{2} + p^{2} T^{4} \) |
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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
−11.16679809343792126919038274667, −10.96555341410425636086309997597, −10.08017141377105238169731944446, −9.581168449578710297688473560356, −9.096303062908669530462883454538, −8.574223034598092966737460294096, −8.149624255490186637541192452864, −7.57660400870567455681765648785, −6.67999960541019473841907537703, −6.13817826308417906229670655342, −5.18812111025287882801104414890, −4.71839188611404283173472155542, −3.75968339734596524308915012701, −2.79882649186254505734959643146, −1.23396492719572603792206010777,
1.23396492719572603792206010777, 2.79882649186254505734959643146, 3.75968339734596524308915012701, 4.71839188611404283173472155542, 5.18812111025287882801104414890, 6.13817826308417906229670655342, 6.67999960541019473841907537703, 7.57660400870567455681765648785, 8.149624255490186637541192452864, 8.574223034598092966737460294096, 9.096303062908669530462883454538, 9.581168449578710297688473560356, 10.08017141377105238169731944446, 10.96555341410425636086309997597, 11.16679809343792126919038274667