L(s) = 1 | + 243·3-s − 1.02e3·4-s + 1.09e4·7-s − 2.48e5·12-s − 1.12e6·13-s − 4.92e6·19-s + 2.65e6·21-s − 9.76e6·25-s − 1.43e7·27-s − 1.11e7·28-s + 4.98e7·31-s + 9.43e7·37-s − 2.72e8·39-s + 4.22e8·43-s − 1.63e8·49-s + 1.14e9·52-s − 1.19e9·57-s + 1.97e8·61-s + 1.07e9·64-s + 1.26e9·67-s − 1.95e9·73-s − 2.37e9·75-s + 5.04e9·76-s − 2.10e9·79-s − 3.48e9·81-s − 2.71e9·84-s − 1.22e10·91-s + ⋯ |
L(s) = 1 | + 3-s − 4-s + 0.648·7-s − 12-s − 3.01·13-s − 1.98·19-s + 0.648·21-s − 25-s − 27-s − 0.648·28-s + 1.74·31-s + 1.36·37-s − 3.01·39-s + 2.87·43-s − 0.578·49-s + 3.01·52-s − 1.98·57-s + 0.233·61-s + 64-s + 0.933·67-s − 0.944·73-s − 75-s + 1.98·76-s − 0.682·79-s − 81-s − 0.648·84-s − 1.95·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+5)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{11}{2})\) |
\(\approx\) |
\(1.066901973\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.066901973\) |
\(L(6)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 3 | $C_2$ | \( 1 - p^{5} T + p^{10} T^{2} \) |
| 7 | $C_2$ | \( 1 - 10907 T + p^{10} T^{2} \) |
good | 2 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 + 560257 T + p^{10} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2024677 T + p^{10} T^{2} )( 1 + 2901574 T + p^{10} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 49326674 T + p^{10} T^{2} )( 1 - 516899 T + p^{10} T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 135214586 T + p^{10} T^{2} )( 1 + 40895593 T + p^{10} T^{2} ) \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 211108739 T + p^{10} T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 1551490727 T + p^{10} T^{2} )( 1 + 1354266001 T + p^{10} T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 2698325411 T + p^{10} T^{2} )( 1 + 1437442918 T + p^{10} T^{2} ) \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2186355743 T + p^{10} T^{2} )( 1 + 4144040686 T + p^{10} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 3959005298 T + p^{10} T^{2} )( 1 + 6059886949 T + p^{10} T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - p^{5} T )^{2}( 1 + p^{5} T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - p^{5} T + p^{10} T^{2} )( 1 + p^{5} T + p^{10} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 884916482 T + p^{10} T^{2} )^{2} \) |
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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
−15.97888810443697348681559611756, −14.97873201255367864012863233165, −14.76065536785130689020871524067, −14.28310877244664970252503517109, −13.68115545707350002583155875086, −12.89041580245344538787327752090, −12.32507367826096577054758094033, −11.54366828828261323572962927404, −10.52178958754663682339631532922, −9.530760343666959544330514427914, −9.437747454675062396311481318086, −8.288036093053852830176556787398, −7.967054357579153138303090384502, −7.03934211646579068904780625138, −5.70590921258513929303530905094, −4.44441158647025928907998056370, −4.41139361766337083007247776931, −2.60052370890738563965330904422, −2.22968747529048781515639301354, −0.39492249496548271021533685501,
0.39492249496548271021533685501, 2.22968747529048781515639301354, 2.60052370890738563965330904422, 4.41139361766337083007247776931, 4.44441158647025928907998056370, 5.70590921258513929303530905094, 7.03934211646579068904780625138, 7.967054357579153138303090384502, 8.288036093053852830176556787398, 9.437747454675062396311481318086, 9.530760343666959544330514427914, 10.52178958754663682339631532922, 11.54366828828261323572962927404, 12.32507367826096577054758094033, 12.89041580245344538787327752090, 13.68115545707350002583155875086, 14.28310877244664970252503517109, 14.76065536785130689020871524067, 14.97873201255367864012863233165, 15.97888810443697348681559611756