| L(s) = 1 | + 2-s − 3-s + 5-s − 6-s − 8-s + 10-s − 11-s − 15-s − 16-s − 22-s + 24-s + 25-s + 27-s + 2·29-s − 30-s − 31-s + 33-s − 40-s + 48-s + 50-s − 53-s + 54-s − 55-s + 2·58-s + 59-s − 62-s + 64-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 5-s − 6-s − 8-s + 10-s − 11-s − 15-s − 16-s − 22-s + 24-s + 25-s + 27-s + 2·29-s − 30-s − 31-s + 33-s − 40-s + 48-s + 50-s − 53-s + 54-s − 55-s + 2·58-s + 59-s − 62-s + 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1382976 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.060862936\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.060862936\) |
| \(L(1)\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 5 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.32110911446975505004905760276, −9.688300914258717542184388682077, −9.635380801605928738865688778126, −8.792415413851405298238500163679, −8.752080930566324294543999380904, −8.110928862295636083794298139221, −7.75365508106348122153428176998, −6.82395877422430105079915811174, −6.78505134034273009792233770489, −6.26781719533747438260107681791, −5.82250174744035564124820305572, −5.39218140143969589184786506432, −5.26509475675830070917360237559, −4.64021122083724918976772870124, −4.45634369964125430872140409869, −3.55651756667587210553152796945, −3.05517299500382483177085281351, −2.59233384005044240956141973490, −1.97540376362818060493153008820, −0.844296705940352217022011758479,
0.844296705940352217022011758479, 1.97540376362818060493153008820, 2.59233384005044240956141973490, 3.05517299500382483177085281351, 3.55651756667587210553152796945, 4.45634369964125430872140409869, 4.64021122083724918976772870124, 5.26509475675830070917360237559, 5.39218140143969589184786506432, 5.82250174744035564124820305572, 6.26781719533747438260107681791, 6.78505134034273009792233770489, 6.82395877422430105079915811174, 7.75365508106348122153428176998, 8.110928862295636083794298139221, 8.752080930566324294543999380904, 8.792415413851405298238500163679, 9.635380801605928738865688778126, 9.688300914258717542184388682077, 10.32110911446975505004905760276
