| L(s) = 1 | + 2-s + 4-s + 8-s − 2·9-s + 4·13-s + 16-s − 2·17-s − 2·18-s + 4·19-s + 6·25-s + 4·26-s + 32-s − 2·34-s − 2·36-s + 4·38-s − 4·43-s − 16·47-s + 2·49-s + 6·50-s + 4·52-s − 12·53-s − 20·59-s + 64-s + 4·67-s − 2·68-s − 2·72-s + 4·76-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.353·8-s − 2/3·9-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.471·18-s + 0.917·19-s + 6/5·25-s + 0.784·26-s + 0.176·32-s − 0.342·34-s − 1/3·36-s + 0.648·38-s − 0.609·43-s − 2.33·47-s + 2/7·49-s + 0.848·50-s + 0.554·52-s − 1.64·53-s − 2.60·59-s + 1/8·64-s + 0.488·67-s − 0.242·68-s − 0.235·72-s + 0.458·76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18496 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.700141892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700141892\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 17 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−11.06353945657785771155648360554, −10.66083065572448283794461045171, −9.880777691483019306956979723285, −9.329859201035597611326728859039, −8.679853076585142412216226314949, −8.219602960383742731107353187015, −7.58895112928270325830249696888, −6.80021997737843946868152154819, −6.32492910092699810719628947849, −5.76336985750473627020290796708, −4.98336492904793365814092736975, −4.48135275103886966381294977281, −3.34376975456505021651134750845, −3.03962686799460235267903434107, −1.61186011784938146153054147471,
1.61186011784938146153054147471, 3.03962686799460235267903434107, 3.34376975456505021651134750845, 4.48135275103886966381294977281, 4.98336492904793365814092736975, 5.76336985750473627020290796708, 6.32492910092699810719628947849, 6.80021997737843946868152154819, 7.58895112928270325830249696888, 8.219602960383742731107353187015, 8.679853076585142412216226314949, 9.329859201035597611326728859039, 9.880777691483019306956979723285, 10.66083065572448283794461045171, 11.06353945657785771155648360554
