| L(s) = 1 | + 20·5-s + 12·7-s − 238·11-s − 472·13-s − 368·17-s − 56·19-s + 1.02e3·23-s − 4.06e3·25-s + 144·29-s − 8.22e3·31-s + 240·35-s − 1.40e4·37-s − 6.69e3·41-s − 1.51e4·43-s + 156·47-s − 3.16e4·49-s − 3.00e3·53-s − 4.76e3·55-s + 2.77e4·59-s − 5.78e4·61-s − 9.44e3·65-s − 7.41e4·67-s + 5.87e4·71-s − 4.96e4·73-s − 2.85e3·77-s − 8.29e4·79-s + 2.55e4·83-s + ⋯ |
| L(s) = 1 | + 0.357·5-s + 0.0925·7-s − 0.593·11-s − 0.774·13-s − 0.308·17-s − 0.0355·19-s + 0.405·23-s − 1.30·25-s + 0.0317·29-s − 1.53·31-s + 0.0331·35-s − 1.69·37-s − 0.622·41-s − 1.24·43-s + 0.0103·47-s − 1.88·49-s − 0.146·53-s − 0.212·55-s + 1.03·59-s − 1.99·61-s − 0.277·65-s − 2.01·67-s + 1.38·71-s − 1.08·73-s − 0.0548·77-s − 1.49·79-s + 0.407·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( 1 - 4 p T + 4469 T^{2} - 4 p^{6} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 12 T + 31769 T^{2} - 12 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 238 T + 65399 T^{2} + 238 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 472 T + 316746 T^{2} + 472 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 368 T + 2392034 T^{2} + 368 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 56 T + 2236818 T^{2} + 56 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1028 T + 11210738 T^{2} - 1028 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 144 T + 27115606 T^{2} - 144 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8228 T + 59283897 T^{2} + 8228 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 14076 T + 173654894 T^{2} + 14076 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 6696 T + 214924702 T^{2} + 6696 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 15112 T + 233547522 T^{2} + 15112 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 156 T + 438351202 T^{2} - 156 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 3004 T + 829164869 T^{2} + 3004 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 27704 T + 1552385318 T^{2} - 27704 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 57856 T + 2525901402 T^{2} + 57856 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 74168 T + 4013167026 T^{2} + 74168 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 58768 T + 3739461902 T^{2} - 58768 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 49606 T + 1887735819 T^{2} + 49606 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 82984 T + 6683867166 T^{2} + 82984 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 25570 T + 6145459415 T^{2} - 25570 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 21480 T - 378921602 T^{2} - 21480 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 79826 T + 17389451667 T^{2} + 79826 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−11.10145065157767210296110161959, −10.76944611737549123897117764318, −10.03106967034424180883576224949, −9.973155749542527735658409470751, −9.129062917198354371185598072747, −8.918182626416143555359766963862, −8.014154132347385018296000132000, −7.85371991160512383897219764947, −6.96062834897083997997721114507, −6.81958518510998512651980683454, −5.87123190617526082569057740676, −5.44317699340944459547565825020, −4.91526027571241224008405259280, −4.28896512849060260825553330367, −3.41451454357046559498352037160, −2.90413662137170426021878356643, −1.88599486029054831176877897060, −1.61769760096525290798613204725, 0, 0,
1.61769760096525290798613204725, 1.88599486029054831176877897060, 2.90413662137170426021878356643, 3.41451454357046559498352037160, 4.28896512849060260825553330367, 4.91526027571241224008405259280, 5.44317699340944459547565825020, 5.87123190617526082569057740676, 6.81958518510998512651980683454, 6.96062834897083997997721114507, 7.85371991160512383897219764947, 8.014154132347385018296000132000, 8.918182626416143555359766963862, 9.129062917198354371185598072747, 9.973155749542527735658409470751, 10.03106967034424180883576224949, 10.76944611737549123897117764318, 11.10145065157767210296110161959
