| L(s) = 1 | − 8·5-s − 294·9-s + 376·13-s + 1.53e3·17-s − 6.20e3·25-s + 5.75e3·29-s − 3.06e4·37-s − 2.44e4·41-s + 2.35e3·45-s − 5.96e3·49-s − 6.68e4·53-s − 6.64e4·61-s − 3.00e3·65-s + 3.78e4·73-s + 2.73e4·81-s − 1.22e4·85-s + 1.55e5·89-s − 1.57e5·97-s − 3.07e4·101-s − 2.86e5·109-s − 1.16e5·113-s − 1.10e5·117-s − 3.00e4·121-s + 7.47e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.143·5-s − 1.20·9-s + 0.617·13-s + 1.28·17-s − 1.98·25-s + 1.27·29-s − 3.68·37-s − 2.26·41-s + 0.173·45-s − 0.354·49-s − 3.26·53-s − 2.28·61-s − 0.0883·65-s + 0.830·73-s + 0.463·81-s − 0.183·85-s + 2.08·89-s − 1.70·97-s − 0.300·101-s − 2.31·109-s − 0.857·113-s − 0.746·117-s − 0.186·121-s + 0.427·125-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 98 p T^{2} + p^{10} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p^{5} T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 5966 T^{2} + p^{10} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 30070 T^{2} + p^{10} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 188 T + p^{5} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 766 T + p^{5} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 4812230 T^{2} + p^{10} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6651886 T^{2} + p^{10} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2876 T + p^{5} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 29445826 T^{2} + p^{10} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 15348 T + p^{5} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 12202 T + p^{5} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 229650614 T^{2} + p^{10} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 47177182 T^{2} + p^{10} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 33412 T + p^{5} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1322710870 T^{2} + p^{10} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 33204 T + p^{5} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 315180634 T^{2} + p^{10} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 2722589134 T^{2} + p^{10} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 18906 T + p^{5} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 6148693790 T^{2} + p^{10} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 386889146 T^{2} + p^{10} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 77866 T + p^{5} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 78930 T + p^{5} T^{2} )^{2} \) |
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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
−10.82540483212220511332924873587, −10.49941427716108776149511649184, −10.04838912246358370531489743147, −9.443514778344111864958246622915, −8.950394432538553067643748003973, −8.440223185230252536080848443244, −7.88427152572932655085286110417, −7.82057202486721003578612372447, −6.66479246032672398944254931718, −6.53423533683134299828981800927, −5.76682132405284826520517116682, −5.28240983499484498289128908566, −4.87259646851450210498735600092, −3.86088388827126595699796466097, −3.29512946766298387106468403779, −3.03560529083943406207371409195, −1.80812758669724149147885485353, −1.43349698000536122783113644606, 0, 0,
1.43349698000536122783113644606, 1.80812758669724149147885485353, 3.03560529083943406207371409195, 3.29512946766298387106468403779, 3.86088388827126595699796466097, 4.87259646851450210498735600092, 5.28240983499484498289128908566, 5.76682132405284826520517116682, 6.53423533683134299828981800927, 6.66479246032672398944254931718, 7.82057202486721003578612372447, 7.88427152572932655085286110417, 8.440223185230252536080848443244, 8.950394432538553067643748003973, 9.443514778344111864958246622915, 10.04838912246358370531489743147, 10.49941427716108776149511649184, 10.82540483212220511332924873587
