| L(s) = 1 | + 3·3-s − 5-s − 4·7-s + 6·9-s + 13-s − 3·15-s − 2·17-s − 16·19-s − 12·21-s − 7·23-s + 9·27-s + 2·29-s − 10·31-s + 4·35-s − 20·37-s + 3·39-s + 6·41-s + 9·43-s − 6·45-s + 6·47-s + 7·49-s − 6·51-s − 2·53-s − 48·57-s + 6·59-s + 61-s − 24·63-s + ⋯ |
| L(s) = 1 | + 1.73·3-s − 0.447·5-s − 1.51·7-s + 2·9-s + 0.277·13-s − 0.774·15-s − 0.485·17-s − 3.67·19-s − 2.61·21-s − 1.45·23-s + 1.73·27-s + 0.371·29-s − 1.79·31-s + 0.676·35-s − 3.28·37-s + 0.480·39-s + 0.937·41-s + 1.37·43-s − 0.894·45-s + 0.875·47-s + 49-s − 0.840·51-s − 0.274·53-s − 6.35·57-s + 0.781·59-s + 0.128·61-s − 3.02·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5475600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4988491921\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4988491921\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 7 T + 26 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 2 T - 25 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 10 T + 69 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 6 T - 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 9 T + 38 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 5 T - 54 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 14 T + 113 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
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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
−9.339067030823599089995902554802, −8.698414667432522331213177464820, −8.589800846416742684248504419469, −8.154314787672656890963035663663, −7.79522363159029733878980032105, −7.09376834894816171592792173836, −7.08815929417576038571256521834, −6.48741011381890084504038595658, −6.32865404376279474599746522359, −5.75027701652718381489635443220, −5.26087852461674338678724791175, −4.40354324947166887430341822868, −4.10830751468055256453102901456, −3.76318210254866977855298770152, −3.71095792962008661279179598282, −2.87937907958528219353259169027, −2.47262792121120660283211694794, −2.05339815909747950766493191248, −1.65197511882814438557174307099, −0.19418922666164051397669138183,
0.19418922666164051397669138183, 1.65197511882814438557174307099, 2.05339815909747950766493191248, 2.47262792121120660283211694794, 2.87937907958528219353259169027, 3.71095792962008661279179598282, 3.76318210254866977855298770152, 4.10830751468055256453102901456, 4.40354324947166887430341822868, 5.26087852461674338678724791175, 5.75027701652718381489635443220, 6.32865404376279474599746522359, 6.48741011381890084504038595658, 7.08815929417576038571256521834, 7.09376834894816171592792173836, 7.79522363159029733878980032105, 8.154314787672656890963035663663, 8.589800846416742684248504419469, 8.698414667432522331213177464820, 9.339067030823599089995902554802
