| L(s) = 1 | − 2-s − 3-s + 6-s − 2·7-s + 8-s + 5·11-s + 2·13-s + 2·14-s − 16-s − 2·17-s + 2·19-s + 2·21-s − 5·22-s − 23-s − 24-s − 2·26-s + 27-s − 5·29-s − 22·31-s − 5·33-s + 2·34-s + 3·37-s − 2·38-s − 2·39-s + 2·41-s − 2·42-s − 11·43-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 0.408·6-s − 0.755·7-s + 0.353·8-s + 1.50·11-s + 0.554·13-s + 0.534·14-s − 1/4·16-s − 0.485·17-s + 0.458·19-s + 0.436·21-s − 1.06·22-s − 0.208·23-s − 0.204·24-s − 0.392·26-s + 0.192·27-s − 0.928·29-s − 3.95·31-s − 0.870·33-s + 0.342·34-s + 0.493·37-s − 0.324·38-s − 0.320·39-s + 0.312·41-s − 0.308·42-s − 1.67·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2385400379\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2385400379\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + 2 T - 3 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + T - 22 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 5 T - 4 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 11 T + 78 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 15 T + 166 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 10 T + 39 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 - 5 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 2 T - 85 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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
−9.585004127726339024595992195419, −9.093699683537551757466144056523, −8.651696765419628105828581767394, −8.348550513080196802249540049065, −7.898853942045239933979178530702, −7.23673052229256695121896308291, −7.07984703539812439955920088129, −6.69264137507242709402436343910, −6.28852033152591719499455088783, −5.87151243690572244600949654251, −5.39795919318608898228975305857, −5.12808268763697669233491459736, −4.40316090061308059444331850455, −3.92609454927725396277777304455, −3.48298723929694611130694529414, −3.34861563014269181423304696061, −2.31057464705248485639998065968, −1.61166838860790509486031164183, −1.39402333390387187584561532926, −0.21472780312589921412621929644,
0.21472780312589921412621929644, 1.39402333390387187584561532926, 1.61166838860790509486031164183, 2.31057464705248485639998065968, 3.34861563014269181423304696061, 3.48298723929694611130694529414, 3.92609454927725396277777304455, 4.40316090061308059444331850455, 5.12808268763697669233491459736, 5.39795919318608898228975305857, 5.87151243690572244600949654251, 6.28852033152591719499455088783, 6.69264137507242709402436343910, 7.07984703539812439955920088129, 7.23673052229256695121896308291, 7.898853942045239933979178530702, 8.348550513080196802249540049065, 8.651696765419628105828581767394, 9.093699683537551757466144056523, 9.585004127726339024595992195419
