| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s + 7-s − 4·8-s + 2·10-s + 3·12-s − 2·14-s − 15-s + 5·16-s − 3·20-s + 21-s + 2·23-s − 4·24-s − 27-s + 3·28-s + 29-s + 2·30-s − 6·32-s − 35-s + 4·40-s + 41-s − 2·42-s − 43-s − 4·46-s + ⋯ |
| L(s) = 1 | − 2·2-s + 3-s + 3·4-s − 5-s − 2·6-s + 7-s − 4·8-s + 2·10-s + 3·12-s − 2·14-s − 15-s + 5·16-s − 3·20-s + 21-s + 2·23-s − 4·24-s − 27-s + 3·28-s + 29-s + 2·30-s − 6·32-s − 35-s + 4·40-s + 41-s − 2·42-s − 43-s − 4·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1587600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5790049545\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5790049545\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 43 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$ | \( ( 1 - T )^{4} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
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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
−9.792943128739386561937933465380, −9.733517709600166635968790034880, −8.949693717962469567784953589874, −8.803774202252009732078253178272, −8.595604449712224635771108269590, −8.115233629884421508509022543006, −7.906275736739085378071031982733, −7.32288600453071522221585173184, −7.18672108008534280345880671456, −6.83026847723848053162463416563, −6.08140068396568642652600271847, −5.62584183176806742642517594738, −5.13723730185795925571389087856, −4.35270252152875581853806743661, −3.86255355668347233751300706722, −3.14053318006425320907809437051, −2.82687086536661439279922764917, −2.33821250662494675158540583083, −1.58125098722236226816032540179, −0.918707549892505776501049893952,
0.918707549892505776501049893952, 1.58125098722236226816032540179, 2.33821250662494675158540583083, 2.82687086536661439279922764917, 3.14053318006425320907809437051, 3.86255355668347233751300706722, 4.35270252152875581853806743661, 5.13723730185795925571389087856, 5.62584183176806742642517594738, 6.08140068396568642652600271847, 6.83026847723848053162463416563, 7.18672108008534280345880671456, 7.32288600453071522221585173184, 7.906275736739085378071031982733, 8.115233629884421508509022543006, 8.595604449712224635771108269590, 8.803774202252009732078253178272, 8.949693717962469567784953589874, 9.733517709600166635968790034880, 9.792943128739386561937933465380
