| L(s) = 1 | + 2-s + 2·4-s + 2·7-s + 5·8-s + 4·11-s − 5·13-s + 2·14-s + 5·16-s − 4·17-s − 8·19-s + 4·22-s − 4·23-s + 5·25-s − 5·26-s + 4·28-s − 8·29-s − 6·31-s + 10·32-s − 4·34-s + 6·37-s − 8·38-s − 12·41-s + 43-s + 8·44-s − 4·46-s − 6·47-s − 11·49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s + 0.755·7-s + 1.76·8-s + 1.20·11-s − 1.38·13-s + 0.534·14-s + 5/4·16-s − 0.970·17-s − 1.83·19-s + 0.852·22-s − 0.834·23-s + 25-s − 0.980·26-s + 0.755·28-s − 1.48·29-s − 1.07·31-s + 1.76·32-s − 0.685·34-s + 0.986·37-s − 1.29·38-s − 1.87·41-s + 0.152·43-s + 1.20·44-s − 0.589·46-s − 0.875·47-s − 1.57·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 29241 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.295425420\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.295425420\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 4 T - 7 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 8 T + 35 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 12 T + 103 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 10 T + 41 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 11 T + 48 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 2 T - 93 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−13.00358723600296357930201328921, −12.72990820154975166673138353569, −11.73795319515122549705207836049, −11.72597041430198767534386264957, −10.99314825236889749223271038992, −10.89319498300486686465672884818, −9.991703005421745185780464941918, −9.770082112684664126323073126712, −8.739122200387657435331285421480, −8.465345632336524695421519797279, −7.63756970877524069298361787844, −7.22863667257789865628661188720, −6.62899897096283456553033020916, −6.30003152929570013923168302982, −5.25237231027046152650155683618, −4.73389840297896153033691600548, −4.26218036196311968434327440702, −3.55767824986701375675868417535, −2.05930157279150413425232928117, −1.98588800280289030643560997342,
1.98588800280289030643560997342, 2.05930157279150413425232928117, 3.55767824986701375675868417535, 4.26218036196311968434327440702, 4.73389840297896153033691600548, 5.25237231027046152650155683618, 6.30003152929570013923168302982, 6.62899897096283456553033020916, 7.22863667257789865628661188720, 7.63756970877524069298361787844, 8.465345632336524695421519797279, 8.739122200387657435331285421480, 9.770082112684664126323073126712, 9.991703005421745185780464941918, 10.89319498300486686465672884818, 10.99314825236889749223271038992, 11.72597041430198767534386264957, 11.73795319515122549705207836049, 12.72990820154975166673138353569, 13.00358723600296357930201328921
