| L(s) = 1 | − 3-s − 4-s + 4·7-s + 9-s + 12-s + 2·13-s − 3·16-s − 6·19-s − 4·21-s − 25-s − 27-s − 4·28-s − 2·31-s − 36-s − 4·37-s − 2·39-s − 10·43-s + 3·48-s + 2·49-s − 2·52-s + 6·57-s − 4·61-s + 4·63-s + 7·64-s − 6·67-s + 75-s + 6·76-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1/2·4-s + 1.51·7-s + 1/3·9-s + 0.288·12-s + 0.554·13-s − 3/4·16-s − 1.37·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 0.755·28-s − 0.359·31-s − 1/6·36-s − 0.657·37-s − 0.320·39-s − 1.52·43-s + 0.433·48-s + 2/7·49-s − 0.277·52-s + 0.794·57-s − 0.512·61-s + 0.503·63-s + 7/8·64-s − 0.733·67-s + 0.115·75-s + 0.688·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243675 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_1$ | \( 1 + T \) |
| 5 | $C_2$ | \( 1 + T^{2} \) |
| 19 | $C_2$ | \( 1 + 6 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 44 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.593917946955164955871663391398, −8.409050662816037013658497211522, −7.85668232398436390657866215570, −7.36340252768348613997453202888, −6.59003416492810382038434874830, −6.53512873780771400495109952978, −5.57075046284373331718582908483, −5.36004521642751391049304360231, −4.62388312892885559329095659142, −4.39223064490415808368029175552, −3.85023417367922247417552763507, −2.92400971955770211327520464920, −1.92577471099237739968377247932, −1.48413346507443369161719444742, 0,
1.48413346507443369161719444742, 1.92577471099237739968377247932, 2.92400971955770211327520464920, 3.85023417367922247417552763507, 4.39223064490415808368029175552, 4.62388312892885559329095659142, 5.36004521642751391049304360231, 5.57075046284373331718582908483, 6.53512873780771400495109952978, 6.59003416492810382038434874830, 7.36340252768348613997453202888, 7.85668232398436390657866215570, 8.409050662816037013658497211522, 8.593917946955164955871663391398
