| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 2·8-s + 10-s − 14-s + 2·16-s − 2·19-s − 20-s + 25-s + 28-s − 31-s − 2·32-s − 35-s + 2·38-s + 2·40-s − 41-s + 2·47-s + 49-s − 50-s − 2·56-s − 59-s + 62-s + 3·64-s − 2·67-s + 70-s + ⋯ |
| L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 2·8-s + 10-s − 14-s + 2·16-s − 2·19-s − 20-s + 25-s + 28-s − 31-s − 2·32-s − 35-s + 2·38-s + 2·40-s − 41-s + 2·47-s + 49-s − 50-s − 2·56-s − 59-s + 62-s + 3·64-s − 2·67-s + 70-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6305121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6305121 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4617739476\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4617739476\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 7 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 19 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
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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
−9.145501754859929174447904801090, −8.736323713411248440203419789786, −8.676405056856754393515236625753, −8.198478000487596485289428798442, −7.81902750449512159214704298116, −7.58846462201935680829343645074, −6.93931555965084014063122329964, −6.89020714323617654750340770777, −6.14274630484919962006846332736, −6.09895773373300085959982092023, −5.33773211109242105518335590082, −5.14344558549566103168630272916, −4.29176497678808437394661167134, −4.19636261136625581228050107981, −3.55015815251004854649158269162, −3.11426226292839122231162671076, −2.49390572277752623463453504994, −2.08705053634266367403091249959, −1.48692283923433881488862561605, −0.53112701819777788176625427430,
0.53112701819777788176625427430, 1.48692283923433881488862561605, 2.08705053634266367403091249959, 2.49390572277752623463453504994, 3.11426226292839122231162671076, 3.55015815251004854649158269162, 4.19636261136625581228050107981, 4.29176497678808437394661167134, 5.14344558549566103168630272916, 5.33773211109242105518335590082, 6.09895773373300085959982092023, 6.14274630484919962006846332736, 6.89020714323617654750340770777, 6.93931555965084014063122329964, 7.58846462201935680829343645074, 7.81902750449512159214704298116, 8.198478000487596485289428798442, 8.676405056856754393515236625753, 8.736323713411248440203419789786, 9.145501754859929174447904801090
