| L(s) = 1 | + 2·2-s + 2·4-s + 4·5-s + 8·10-s − 2·13-s − 4·16-s − 6·17-s + 8·20-s + 11·25-s − 4·26-s − 8·32-s − 12·34-s − 14·37-s + 16·41-s + 22·50-s − 4·52-s − 18·53-s + 24·61-s − 8·64-s − 8·65-s − 12·68-s − 22·73-s − 28·74-s − 16·80-s + 32·82-s − 24·85-s + 26·97-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 1.78·5-s + 2.52·10-s − 0.554·13-s − 16-s − 1.45·17-s + 1.78·20-s + 11/5·25-s − 0.784·26-s − 1.41·32-s − 2.05·34-s − 2.30·37-s + 2.49·41-s + 3.11·50-s − 0.554·52-s − 2.47·53-s + 3.07·61-s − 64-s − 0.992·65-s − 1.45·68-s − 2.57·73-s − 3.25·74-s − 1.78·80-s + 3.53·82-s − 2.60·85-s + 2.63·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.065142573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.065142573\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 18 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 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
−12.99030959944853555289996114351, −12.69794170135484687304413895935, −12.18370566131111394162278799771, −11.49029736143249133136551533241, −11.02191249766749448320241458224, −10.57247539133846836722364304916, −9.870320758800327287005653304896, −9.555475699564994929062637634935, −8.758274899783008978368742428221, −8.739973580780035675871218399526, −7.46075314863325815153746791658, −6.92631001022100348507174626909, −6.37224151113724282744104274704, −5.96147912093056451593267216430, −5.36059816736116392293212599542, −4.84360594018672742592642180007, −4.29322535774556801723332538839, −3.29130633826925225407131880272, −2.48247368747939277244217051032, −1.92433628645762413952817508959,
1.92433628645762413952817508959, 2.48247368747939277244217051032, 3.29130633826925225407131880272, 4.29322535774556801723332538839, 4.84360594018672742592642180007, 5.36059816736116392293212599542, 5.96147912093056451593267216430, 6.37224151113724282744104274704, 6.92631001022100348507174626909, 7.46075314863325815153746791658, 8.739973580780035675871218399526, 8.758274899783008978368742428221, 9.555475699564994929062637634935, 9.870320758800327287005653304896, 10.57247539133846836722364304916, 11.02191249766749448320241458224, 11.49029736143249133136551533241, 12.18370566131111394162278799771, 12.69794170135484687304413895935, 12.99030959944853555289996114351
