| L(s) = 1 | + 2-s + 2·5-s + 4·7-s + 8-s − 2·9-s + 2·10-s − 2·13-s + 4·14-s − 16-s − 17-s − 2·18-s − 2·19-s − 6·23-s + 2·25-s − 2·26-s − 4·29-s − 6·31-s − 6·32-s − 34-s + 8·35-s + 3·37-s − 2·38-s + 2·40-s + 2·41-s − 4·43-s − 4·45-s − 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s + 1.51·7-s + 0.353·8-s − 2/3·9-s + 0.632·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 0.242·17-s − 0.471·18-s − 0.458·19-s − 1.25·23-s + 2/5·25-s − 0.392·26-s − 0.742·29-s − 1.07·31-s − 1.06·32-s − 0.171·34-s + 1.35·35-s + 0.493·37-s − 0.324·38-s + 0.316·40-s + 0.312·41-s − 0.609·43-s − 0.596·45-s − 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19385 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19385 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.882467565\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.882467565\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 3 T + p T^{2} ) \) |
| 3877 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 98 T + p T^{2} ) \) |
| good | 2 | $D_{4}$ | \( 1 - T + T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 7 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $D_{4}$ | \( 1 + T - 14 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 21 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 43 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 3 T + 40 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 2 T - 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 4 T + 53 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 12 T + 82 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $D_{4}$ | \( 1 - 12 T + 106 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 23 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 6 T + 133 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 7 T + 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T - 102 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 40 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 4 T - 35 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 3 T + 82 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
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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
−15.7972047496, −14.9159289998, −14.7498896263, −14.2905658638, −13.9287315291, −13.5974212612, −12.9631404899, −12.5656707389, −11.9118806126, −11.3486451114, −11.0317200629, −10.5156521971, −9.85826678243, −9.25097559909, −8.74802367039, −8.10254239982, −7.62129453274, −6.93651364483, −6.14601273177, −5.43456155106, −5.18412202111, −4.38531195674, −3.81849934369, −2.37762571607, −1.88424509687,
1.88424509687, 2.37762571607, 3.81849934369, 4.38531195674, 5.18412202111, 5.43456155106, 6.14601273177, 6.93651364483, 7.62129453274, 8.10254239982, 8.74802367039, 9.25097559909, 9.85826678243, 10.5156521971, 11.0317200629, 11.3486451114, 11.9118806126, 12.5656707389, 12.9631404899, 13.5974212612, 13.9287315291, 14.2905658638, 14.7498896263, 14.9159289998, 15.7972047496
