L(s) = 1 | + 3·3-s − 5-s + 3·9-s + 5·11-s + 6·13-s − 3·15-s − 17-s + 6·19-s − 6·23-s − 18·29-s − 4·31-s + 15·33-s − 2·37-s + 18·39-s + 8·41-s + 20·43-s − 3·45-s − 47-s − 3·51-s − 4·53-s − 5·55-s + 18·57-s − 8·59-s − 8·61-s − 6·65-s − 12·67-s − 18·69-s + ⋯ |
L(s) = 1 | + 1.73·3-s − 0.447·5-s + 9-s + 1.50·11-s + 1.66·13-s − 0.774·15-s − 0.242·17-s + 1.37·19-s − 1.25·23-s − 3.34·29-s − 0.718·31-s + 2.61·33-s − 0.328·37-s + 2.88·39-s + 1.24·41-s + 3.04·43-s − 0.447·45-s − 0.145·47-s − 0.420·51-s − 0.549·53-s − 0.674·55-s + 2.38·57-s − 1.04·59-s − 1.02·61-s − 0.744·65-s − 1.46·67-s − 2.16·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.136781374\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.136781374\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T - 16 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 6 T + 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T - 46 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 4 T - 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 8 T + 5 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−10.07874805288484003330024534610, −9.314339183736443234315971302491, −9.170036079240335566035548870042, −9.144391342023484309781584764377, −8.780991723512763655898270630825, −7.984498717222100246955147740029, −7.84282940941575329803294748095, −7.39754411231380091219373760824, −7.18652279334543349240355008413, −6.22680407899718166068467139003, −5.85384386671571582830167338039, −5.82795065963223808062270303981, −4.75357227039929445424942766486, −3.99431508679137193633347955316, −3.95417231387729679149553435415, −3.29885871641974753156334332248, −3.23561328476068011249769120887, −2.06012136757807741944124969827, −1.86184011198517421290437146900, −0.893051422165929279927283314065,
0.893051422165929279927283314065, 1.86184011198517421290437146900, 2.06012136757807741944124969827, 3.23561328476068011249769120887, 3.29885871641974753156334332248, 3.95417231387729679149553435415, 3.99431508679137193633347955316, 4.75357227039929445424942766486, 5.82795065963223808062270303981, 5.85384386671571582830167338039, 6.22680407899718166068467139003, 7.18652279334543349240355008413, 7.39754411231380091219373760824, 7.84282940941575329803294748095, 7.984498717222100246955147740029, 8.780991723512763655898270630825, 9.144391342023484309781584764377, 9.170036079240335566035548870042, 9.314339183736443234315971302491, 10.07874805288484003330024534610