| L(s) = 1 | + 2-s + 2·4-s + 3·5-s − 4·7-s + 5·8-s + 3·10-s + 6·11-s + 2·13-s − 4·14-s + 5·16-s − 17-s + 4·19-s + 6·20-s + 6·22-s + 4·23-s + 5·25-s + 2·26-s − 8·28-s + 14·29-s − 7·31-s + 10·32-s − 34-s − 12·35-s − 8·37-s + 4·38-s + 15·40-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 1.34·5-s − 1.51·7-s + 1.76·8-s + 0.948·10-s + 1.80·11-s + 0.554·13-s − 1.06·14-s + 5/4·16-s − 0.242·17-s + 0.917·19-s + 1.34·20-s + 1.27·22-s + 0.834·23-s + 25-s + 0.392·26-s − 1.51·28-s + 2.59·29-s − 1.25·31-s + 1.76·32-s − 0.171·34-s − 2.02·35-s − 1.31·37-s + 0.648·38-s + 2.37·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1147041 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.147032900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.147032900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 17 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 8 T + 27 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 7 T + 2 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 5 T - 34 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−10.12237919082952657319622433503, −9.767731668144289755730142869259, −9.257128849302038995483283601708, −9.141596268552397601724739856327, −8.393056621694735950159986182568, −8.165829775413167881415288830911, −7.11728869835905466669126672404, −6.99493873357664471222860795373, −6.59957024545222033655778502661, −6.47281252993319693230881983875, −5.98594335069302067964830398988, −5.33767484341052500650891695341, −5.00556484727790607526042897580, −4.47933589507849181220015900441, −3.65890322653304908679200282535, −3.48383452675271022971091152555, −2.90577556172763788980181151164, −2.22103824551955356977735315487, −1.49919855892003166702434593954, −1.15807201173852546747242214322,
1.15807201173852546747242214322, 1.49919855892003166702434593954, 2.22103824551955356977735315487, 2.90577556172763788980181151164, 3.48383452675271022971091152555, 3.65890322653304908679200282535, 4.47933589507849181220015900441, 5.00556484727790607526042897580, 5.33767484341052500650891695341, 5.98594335069302067964830398988, 6.47281252993319693230881983875, 6.59957024545222033655778502661, 6.99493873357664471222860795373, 7.11728869835905466669126672404, 8.165829775413167881415288830911, 8.393056621694735950159986182568, 9.141596268552397601724739856327, 9.257128849302038995483283601708, 9.767731668144289755730142869259, 10.12237919082952657319622433503
