| L(s) = 1 | + 2-s + 3-s + 6-s − 8-s + 3·9-s + 11-s + 2·13-s − 16-s − 6·17-s + 3·18-s + 2·19-s + 22-s + 6·23-s − 24-s + 5·25-s + 2·26-s + 8·27-s + 18·29-s − 4·31-s + 33-s − 6·34-s − 2·37-s + 2·38-s + 2·39-s + 12·41-s − 8·43-s + 6·46-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s − 0.353·8-s + 9-s + 0.301·11-s + 0.554·13-s − 1/4·16-s − 1.45·17-s + 0.707·18-s + 0.458·19-s + 0.213·22-s + 1.25·23-s − 0.204·24-s + 25-s + 0.392·26-s + 1.53·27-s + 3.34·29-s − 0.718·31-s + 0.174·33-s − 1.02·34-s − 0.328·37-s + 0.324·38-s + 0.320·39-s + 1.87·41-s − 1.21·43-s + 0.884·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1162084 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.527695485\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.527695485\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T - 69 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 5 T - 54 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.08415165423804370100830699077, −9.587159948076271020332804923471, −9.255661462487002235834745161795, −8.748428681230517740595970847040, −8.408241477529121046090652651866, −8.347297881760747658864483890905, −7.44747866639599614575568703753, −7.01948105860642191046275300789, −6.72806210687619591578597191674, −6.40874128885435513045069454441, −5.91832664086939168897531281991, −4.98380498282920744647575634358, −4.88251649371443125774688056225, −4.49004701694734968738406571790, −3.96281355148462004812862051896, −3.37129338803553429510955927136, −2.79874522381940100093547824940, −2.54536368458609811717449923553, −1.45698573322672188167741303031, −0.926571860457696678279022541406,
0.926571860457696678279022541406, 1.45698573322672188167741303031, 2.54536368458609811717449923553, 2.79874522381940100093547824940, 3.37129338803553429510955927136, 3.96281355148462004812862051896, 4.49004701694734968738406571790, 4.88251649371443125774688056225, 4.98380498282920744647575634358, 5.91832664086939168897531281991, 6.40874128885435513045069454441, 6.72806210687619591578597191674, 7.01948105860642191046275300789, 7.44747866639599614575568703753, 8.347297881760747658864483890905, 8.408241477529121046090652651866, 8.748428681230517740595970847040, 9.255661462487002235834745161795, 9.587159948076271020332804923471, 10.08415165423804370100830699077
