| L(s) = 1 | + 6·3-s + 484·7-s − 693·9-s − 5.23e3·13-s − 1.15e4·19-s + 2.90e3·21-s + 5.33e3·25-s − 8.53e3·27-s − 4.08e4·31-s + 9.35e4·37-s − 3.14e4·39-s − 1.37e5·43-s − 5.96e4·49-s − 6.94e4·57-s − 4.95e4·61-s − 3.35e5·63-s + 1.68e5·67-s − 2.27e5·73-s + 3.19e4·75-s − 3.19e5·79-s + 4.54e5·81-s − 2.53e6·91-s − 2.45e5·93-s + 1.79e6·97-s − 3.19e6·103-s − 4.43e6·109-s + 5.61e5·111-s + ⋯ |
| L(s) = 1 | + 2/9·3-s + 1.41·7-s − 0.950·9-s − 2.38·13-s − 1.68·19-s + 0.313·21-s + 0.341·25-s − 0.433·27-s − 1.37·31-s + 1.84·37-s − 0.529·39-s − 1.72·43-s − 0.506·49-s − 0.374·57-s − 0.218·61-s − 1.34·63-s + 0.560·67-s − 0.585·73-s + 0.0758·75-s − 0.647·79-s + 0.854·81-s − 3.36·91-s − 0.305·93-s + 1.97·97-s − 2.92·103-s − 3.42·109-s + 0.410·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 36864 ^{s/2} \, \Gamma_{\C}(s+3)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.1317484294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1317484294\) |
| \(L(4)\) |
|
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 | $C_2$ | \( 1 - 2 p T + p^{6} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 1066 p T^{2} + p^{12} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 242 T + p^{6} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 3362 p^{2} T^{2} + p^{12} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2618 T + p^{6} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 1905982 T^{2} + p^{12} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 5786 T + p^{6} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 208876898 T^{2} + p^{12} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 1035966962 T^{2} + p^{12} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 20446 T + p^{6} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 46774 T + p^{6} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 9487663202 T^{2} + p^{12} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 68618 T + p^{6} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 21106800578 T^{2} + p^{12} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 14819087378 T^{2} + p^{12} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 61991044562 T^{2} + p^{12} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 24794 T + p^{6} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 84358 T + p^{6} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 151063967522 T^{2} + p^{12} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 113806 T + p^{6} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 159742 T + p^{6} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 388294032818 T^{2} + p^{12} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 583819025758 T^{2} + p^{12} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 899522 T + p^{6} T^{2} )^{2} \) |
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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
−12.21125437366542670086421123071, −11.25355753365075487374780782889, −10.79269557375280648196141387059, −10.29809272158155888467244760609, −9.475887337537951396765104910533, −9.405690671078241324779696979593, −8.453097966960030547901703692282, −8.221919568117945181470444905874, −7.77093632365955633590152212565, −7.14596694207534161819043859341, −6.61996112057951753453366160478, −5.81478284743875980186674082107, −5.22764326492750383587116039517, −4.72310406514782775628104007711, −4.35126021986355651075096031869, −3.35043079950084179666767309945, −2.47006725858960065088936464252, −2.20075633765445341389679004641, −1.39518886382477515473785486119, −0.093913167476081235665674802087,
0.093913167476081235665674802087, 1.39518886382477515473785486119, 2.20075633765445341389679004641, 2.47006725858960065088936464252, 3.35043079950084179666767309945, 4.35126021986355651075096031869, 4.72310406514782775628104007711, 5.22764326492750383587116039517, 5.81478284743875980186674082107, 6.61996112057951753453366160478, 7.14596694207534161819043859341, 7.77093632365955633590152212565, 8.221919568117945181470444905874, 8.453097966960030547901703692282, 9.405690671078241324779696979593, 9.475887337537951396765104910533, 10.29809272158155888467244760609, 10.79269557375280648196141387059, 11.25355753365075487374780782889, 12.21125437366542670086421123071
