| L(s) = 1 | + 4·5-s − 12·13-s − 4·17-s + 2·25-s + 16·29-s − 12·37-s + 20·41-s + 49-s − 8·53-s − 4·61-s − 48·65-s − 4·73-s − 16·85-s − 28·89-s − 4·97-s − 36·101-s − 28·109-s + 14·121-s − 28·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + 149-s + 151-s + 157-s + ⋯ |
| L(s) = 1 | + 1.78·5-s − 3.32·13-s − 0.970·17-s + 2/5·25-s + 2.97·29-s − 1.97·37-s + 3.12·41-s + 1/7·49-s − 1.09·53-s − 0.512·61-s − 5.95·65-s − 0.468·73-s − 1.73·85-s − 2.96·89-s − 0.406·97-s − 3.58·101-s − 2.68·109-s + 1.27·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 508032 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−8.315712661651651867252370030676, −7.81574295644308953149169772105, −7.10199896363463577446945617508, −7.06349862549190184985368579025, −6.40338470798410886609758690223, −6.01034584523872431338499373875, −5.26501418343618196652528930650, −5.23049644801577170840071990344, −4.39604700645425257403950589550, −4.31339197807007275668048232269, −2.76055910891770182215747620528, −2.73183663384879672157762604102, −2.21321859651013231545979667053, −1.45089519141154609279134539659, 0,
1.45089519141154609279134539659, 2.21321859651013231545979667053, 2.73183663384879672157762604102, 2.76055910891770182215747620528, 4.31339197807007275668048232269, 4.39604700645425257403950589550, 5.23049644801577170840071990344, 5.26501418343618196652528930650, 6.01034584523872431338499373875, 6.40338470798410886609758690223, 7.06349862549190184985368579025, 7.10199896363463577446945617508, 7.81574295644308953149169772105, 8.315712661651651867252370030676
