L(s) = 1 | + 4·5-s + 9-s + 4·13-s + 4·17-s + 2·25-s − 12·29-s − 12·37-s − 12·41-s + 4·45-s − 14·49-s + 4·53-s + 4·61-s + 16·65-s + 20·73-s + 81-s + 16·85-s − 12·89-s + 4·97-s + 36·101-s + 4·109-s + 36·113-s + 4·117-s − 6·121-s − 28·125-s + 127-s + 131-s + 137-s + ⋯ |
L(s) = 1 | + 1.78·5-s + 1/3·9-s + 1.10·13-s + 0.970·17-s + 2/5·25-s − 2.22·29-s − 1.97·37-s − 1.87·41-s + 0.596·45-s − 2·49-s + 0.549·53-s + 0.512·61-s + 1.98·65-s + 2.34·73-s + 1/9·81-s + 1.73·85-s − 1.27·89-s + 0.406·97-s + 3.58·101-s + 0.383·109-s + 3.38·113-s + 0.369·117-s − 0.545·121-s − 2.50·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.817673508\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.817673508\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 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 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−10.20677110069933956172337352464, −9.785656483951954927896383611288, −9.549120050570062839357860330869, −8.777512613188475551383845278968, −8.456614023795656741730916500766, −7.61947548271568794045481490015, −7.13146007277855219299925953915, −6.21040581024747445781537308376, −6.16826675668887144123811676692, −5.24179959934618595033608799542, −5.14444399128618341481533912587, −3.71523857372513194753595622108, −3.43299575130251299450107440988, −1.99999348241168201324552330088, −1.64229196717434907696557687175,
1.64229196717434907696557687175, 1.99999348241168201324552330088, 3.43299575130251299450107440988, 3.71523857372513194753595622108, 5.14444399128618341481533912587, 5.24179959934618595033608799542, 6.16826675668887144123811676692, 6.21040581024747445781537308376, 7.13146007277855219299925953915, 7.61947548271568794045481490015, 8.456614023795656741730916500766, 8.777512613188475551383845278968, 9.549120050570062839357860330869, 9.785656483951954927896383611288, 10.20677110069933956172337352464