| L(s) = 1 | − 2-s − 2·3-s − 4-s + 2·6-s + 3·8-s + 9-s − 10·11-s + 2·12-s + 2·13-s − 16-s − 18-s + 10·22-s − 18·23-s − 6·24-s − 25-s − 2·26-s + 4·27-s − 5·32-s + 20·33-s − 36-s + 16·37-s − 4·39-s + 10·44-s + 18·46-s + 8·47-s + 2·48-s − 13·49-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.15·3-s − 1/2·4-s + 0.816·6-s + 1.06·8-s + 1/3·9-s − 3.01·11-s + 0.577·12-s + 0.554·13-s − 1/4·16-s − 0.235·18-s + 2.13·22-s − 3.75·23-s − 1.22·24-s − 1/5·25-s − 0.392·26-s + 0.769·27-s − 0.883·32-s + 3.48·33-s − 1/6·36-s + 2.63·37-s − 0.640·39-s + 1.50·44-s + 2.65·46-s + 1.16·47-s + 0.288·48-s − 1.85·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 535824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 535824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + T + p T^{2} \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 61 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.082044785037915401605480045323, −7.66039763333771123227596439320, −7.46804311358618221432298100574, −6.50765477770331403909843587559, −6.04657779065908234361333580545, −5.60157612088078313975836309422, −5.46082075171763164201344547261, −4.76483304272257054947779033705, −4.25666915587159511841100490901, −3.87537067567035553055055523309, −2.69052546193369796226312790874, −2.42195587063124416020477834009, −1.35631448215761706727244320677, 0, 0,
1.35631448215761706727244320677, 2.42195587063124416020477834009, 2.69052546193369796226312790874, 3.87537067567035553055055523309, 4.25666915587159511841100490901, 4.76483304272257054947779033705, 5.46082075171763164201344547261, 5.60157612088078313975836309422, 6.04657779065908234361333580545, 6.50765477770331403909843587559, 7.46804311358618221432298100574, 7.66039763333771123227596439320, 8.082044785037915401605480045323
