| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s − 2·9-s + 12-s + 16-s − 2·18-s + 2·19-s − 6·23-s + 24-s − 10·25-s − 5·27-s − 18·29-s + 32-s − 2·36-s + 2·38-s + 16·43-s − 6·46-s + 48-s − 13·49-s − 10·50-s + 6·53-s − 5·54-s + 2·57-s − 18·58-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s − 2/3·9-s + 0.288·12-s + 1/4·16-s − 0.471·18-s + 0.458·19-s − 1.25·23-s + 0.204·24-s − 2·25-s − 0.962·27-s − 3.34·29-s + 0.176·32-s − 1/3·36-s + 0.324·38-s + 2.43·43-s − 0.884·46-s + 0.144·48-s − 1.85·49-s − 1.41·50-s + 0.824·53-s − 0.680·54-s + 0.264·57-s − 2.36·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 415872 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.165660732182309128185574185775, −7.944498198632498175704416365123, −7.53492665725128185091116505938, −7.14449686564876148233582580493, −6.40690082764165790518888770884, −5.85342474444132611353121181432, −5.56559135650201084358170066267, −5.32600053322851104543262796765, −4.18290710455057630899406360415, −3.98931312476793193392332292270, −3.56295849457415925614690839234, −2.80468657266648836263703727877, −2.16004950277651645615210004057, −1.70600515778630825145695828253, 0,
1.70600515778630825145695828253, 2.16004950277651645615210004057, 2.80468657266648836263703727877, 3.56295849457415925614690839234, 3.98931312476793193392332292270, 4.18290710455057630899406360415, 5.32600053322851104543262796765, 5.56559135650201084358170066267, 5.85342474444132611353121181432, 6.40690082764165790518888770884, 7.14449686564876148233582580493, 7.53492665725128185091116505938, 7.944498198632498175704416365123, 8.165660732182309128185574185775
