| L(s) = 1 | − 2-s + 3-s + 4-s − 6-s − 8-s + 9-s − 6·11-s + 12-s + 16-s − 9·17-s − 18-s + 19-s + 6·22-s − 24-s + 2·25-s + 27-s − 32-s − 6·33-s + 9·34-s + 36-s − 38-s − 3·41-s − 2·43-s − 6·44-s + 48-s − 4·49-s − 2·50-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1/2·4-s − 0.408·6-s − 0.353·8-s + 1/3·9-s − 1.80·11-s + 0.288·12-s + 1/4·16-s − 2.18·17-s − 0.235·18-s + 0.229·19-s + 1.27·22-s − 0.204·24-s + 2/5·25-s + 0.192·27-s − 0.176·32-s − 1.04·33-s + 1.54·34-s + 1/6·36-s − 0.162·38-s − 0.468·41-s − 0.304·43-s − 0.904·44-s + 0.144·48-s − 4/7·49-s − 0.282·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93312 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | $C_1$ | \( 1 + T \) |
| 3 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 19 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 61 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 89 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 103 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + 10 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
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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
−9.238097176676544541856919447609, −8.755320776572624605657268556796, −8.589031677461357649762066369538, −7.86660308641488423632513423103, −7.55904839014797635084846447901, −6.97991898123414951150681433901, −6.49242845094790160210673561057, −5.85346497436345917136603741245, −5.07562986357202465204427362014, −4.66813422565207745216710594028, −3.87488613206077017862725546029, −2.89165547258907086481720434289, −2.55542664503741642179895212946, −1.71012766301984892032160078976, 0,
1.71012766301984892032160078976, 2.55542664503741642179895212946, 2.89165547258907086481720434289, 3.87488613206077017862725546029, 4.66813422565207745216710594028, 5.07562986357202465204427362014, 5.85346497436345917136603741245, 6.49242845094790160210673561057, 6.97991898123414951150681433901, 7.55904839014797635084846447901, 7.86660308641488423632513423103, 8.589031677461357649762066369538, 8.755320776572624605657268556796, 9.238097176676544541856919447609
