| L(s) = 1 | − 2-s − 5-s + 8-s − 3·9-s + 10-s + 2·11-s − 2·13-s − 16-s + 3·18-s − 14·19-s − 2·22-s − 3·23-s + 5·25-s + 2·26-s + 8·29-s − 4·31-s − 12·37-s + 14·38-s − 40-s + 12·41-s + 8·43-s + 3·45-s + 3·46-s + 8·47-s − 5·50-s + 8·53-s − 2·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.447·5-s + 0.353·8-s − 9-s + 0.316·10-s + 0.603·11-s − 0.554·13-s − 1/4·16-s + 0.707·18-s − 3.21·19-s − 0.426·22-s − 0.625·23-s + 25-s + 0.392·26-s + 1.48·29-s − 0.718·31-s − 1.97·37-s + 2.27·38-s − 0.158·40-s + 1.87·41-s + 1.21·43-s + 0.447·45-s + 0.442·46-s + 1.16·47-s − 0.707·50-s + 1.09·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777924 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4413487979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4413487979\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T + 35 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 12 T + 103 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 8 T + 17 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 12 T + 61 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 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
−10.51696023103365853327884837393, −10.13833456294304709366078469746, −9.172582569827637244272936447534, −8.938362876323731976898734086150, −8.723106176718281520329632477332, −8.586182597597517043864144366873, −7.73807044408705986066664800566, −7.57317342490977563656494597802, −7.04055208374458847289762896191, −6.40291338645449863062726797109, −6.13943128522560623072417552031, −5.76679722819547495800601514875, −4.80737181645318555065345541655, −4.62247828969651111851188165476, −4.02866685332688906104050863501, −3.60009953889248425198930802688, −2.54644472288887540353028197812, −2.46001900906047846564147384262, −1.47366099525737588514248138513, −0.36283678923849524452357290607,
0.36283678923849524452357290607, 1.47366099525737588514248138513, 2.46001900906047846564147384262, 2.54644472288887540353028197812, 3.60009953889248425198930802688, 4.02866685332688906104050863501, 4.62247828969651111851188165476, 4.80737181645318555065345541655, 5.76679722819547495800601514875, 6.13943128522560623072417552031, 6.40291338645449863062726797109, 7.04055208374458847289762896191, 7.57317342490977563656494597802, 7.73807044408705986066664800566, 8.586182597597517043864144366873, 8.723106176718281520329632477332, 8.938362876323731976898734086150, 9.172582569827637244272936447534, 10.13833456294304709366078469746, 10.51696023103365853327884837393
