| L(s) = 1 | − 4·2-s + 6·3-s + 14·4-s − 30·5-s − 24·6-s + 24·8-s − 75·9-s + 120·10-s − 58·11-s + 84·12-s − 180·15-s − 96·16-s − 246·17-s + 300·18-s + 642·19-s − 420·20-s + 232·22-s + 290·23-s + 144·24-s − 461·25-s − 522·27-s − 2.17e3·29-s + 720·30-s + 3.61e3·31-s + 1.23e3·32-s − 348·33-s + 984·34-s + ⋯ |
| L(s) = 1 | − 2-s + 2/3·3-s + 7/8·4-s − 6/5·5-s − 2/3·6-s + 3/8·8-s − 0.925·9-s + 6/5·10-s − 0.479·11-s + 7/12·12-s − 4/5·15-s − 3/8·16-s − 0.851·17-s + 0.925·18-s + 1.77·19-s − 1.04·20-s + 0.479·22-s + 0.548·23-s + 1/4·24-s − 0.737·25-s − 0.716·27-s − 2.58·29-s + 4/5·30-s + 3.76·31-s + 1.20·32-s − 0.319·33-s + 0.851·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2401 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.5077499473\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5077499473\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2^2$ | \( 1 + 10 p^{2} T^{2} + p^{8} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 + p^{2} T + p T^{2} - 9 p^{3} T^{3} - 79 p^{2} T^{4} - 9 p^{7} T^{5} + p^{9} T^{6} + p^{14} T^{7} + p^{16} T^{8} \) |
| 3 | $D_4\times C_2$ | \( 1 - 2 p T + 37 p T^{2} - 22 p^{3} T^{3} + 52 p^{4} T^{4} - 22 p^{7} T^{5} + 37 p^{9} T^{6} - 2 p^{13} T^{7} + p^{16} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 6 p T + 1361 T^{2} + 6366 p T^{3} + 922596 T^{4} + 6366 p^{5} T^{5} + 1361 p^{8} T^{6} + 6 p^{13} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 58 T - 20401 T^{2} - 319986 T^{3} + 301164020 T^{4} - 319986 p^{4} T^{5} - 20401 p^{8} T^{6} + 58 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 58972 T^{2} + 1740248838 T^{4} - 58972 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 246 T + 115961 T^{2} + 23564094 T^{3} + 3884560692 T^{4} + 23564094 p^{4} T^{5} + 115961 p^{8} T^{6} + 246 p^{12} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 642 T + 342023 T^{2} - 131375670 T^{3} + 42796461732 T^{4} - 131375670 p^{4} T^{5} + 342023 p^{8} T^{6} - 642 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 290 T - 459625 T^{2} + 4627530 T^{3} + 193791262244 T^{4} + 4627530 p^{4} T^{5} - 459625 p^{8} T^{6} - 290 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 1088 T + 1602698 T^{2} + 1088 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 3618 T + 7245671 T^{2} - 10428389334 T^{3} + 11484731993796 T^{4} - 10428389334 p^{4} T^{5} + 7245671 p^{8} T^{6} - 3618 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 270 T - 3455695 T^{2} - 59326290 T^{3} + 8801876883204 T^{4} - 59326290 p^{4} T^{5} - 3455695 p^{8} T^{6} + 270 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 7249372 T^{2} + 28218527830086 T^{4} - 7249372 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1236 T + 4524526 T^{2} - 1236 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 1542 T + 8442143 T^{2} + 11795613810 T^{3} + 38571981640692 T^{4} + 11795613810 p^{4} T^{5} + 8442143 p^{8} T^{6} + 1542 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 4510 T + 2017313 T^{2} + 11463630750 T^{3} + 112971660447908 T^{4} + 11463630750 p^{4} T^{5} + 2017313 p^{8} T^{6} + 4510 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 2526 T + 13587671 T^{2} - 28949927754 T^{3} + 10291335854532 T^{4} - 28949927754 p^{4} T^{5} + 13587671 p^{8} T^{6} - 2526 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 282 T + 23923217 T^{2} + 6738871938 T^{3} + 379712413586628 T^{4} + 6738871938 p^{4} T^{5} + 23923217 p^{8} T^{6} + 282 p^{12} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 1318 T - 38954849 T^{2} + 513665458 T^{3} + 1214763993160084 T^{4} + 513665458 p^{4} T^{5} - 38954849 p^{8} T^{6} + 1318 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 5204 T + 56347598 T^{2} + 5204 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 5214 T + 63240953 T^{2} - 282489415494 T^{3} + 2386249153485972 T^{4} - 282489415494 p^{4} T^{5} + 63240953 p^{8} T^{6} - 5214 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 8110 T - 25860665 T^{2} + 111371410330 T^{3} + 4317626189098564 T^{4} + 111371410330 p^{4} T^{5} - 25860665 p^{8} T^{6} + 8110 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 166506148 T^{2} + 11414799577931910 T^{4} - 166506148 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 33990 T + 597537041 T^{2} - 7220507290590 T^{3} + 65352518353788900 T^{4} - 7220507290590 p^{4} T^{5} + 597537041 p^{8} T^{6} - 33990 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 137047516 T^{2} + 19765432645146054 T^{4} - 137047516 p^{8} T^{6} + p^{16} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−16.91753668168640460051648153884, −16.24072330190243185278625729937, −16.20697775890283475785899555058, −15.60934794768144622514547500379, −15.50514326345098503640365906933, −14.79168708061966432103980656434, −14.65440519442219863039888674072, −13.74224001808230354833033917117, −13.45744470465150011139308882123, −13.44169416177175947844631363405, −12.31923759615994163169820999040, −11.82026904593738246537811358404, −11.64038034658922173593467619244, −10.96948430456323517638989910064, −10.80866556962478702608479678557, −9.778335567515717376235668356354, −9.652166542892594551482947676731, −8.868463082374881474098087804275, −8.135322344559972522236817182486, −7.889749218084687620260149019359, −7.58222498562621002005891816579, −6.69324740543004824973130781459, −5.62660667520476570351731370592, −4.41042581073859805023009197156, −3.01511646925818929048655556808,
3.01511646925818929048655556808, 4.41042581073859805023009197156, 5.62660667520476570351731370592, 6.69324740543004824973130781459, 7.58222498562621002005891816579, 7.889749218084687620260149019359, 8.135322344559972522236817182486, 8.868463082374881474098087804275, 9.652166542892594551482947676731, 9.778335567515717376235668356354, 10.80866556962478702608479678557, 10.96948430456323517638989910064, 11.64038034658922173593467619244, 11.82026904593738246537811358404, 12.31923759615994163169820999040, 13.44169416177175947844631363405, 13.45744470465150011139308882123, 13.74224001808230354833033917117, 14.65440519442219863039888674072, 14.79168708061966432103980656434, 15.50514326345098503640365906933, 15.60934794768144622514547500379, 16.20697775890283475785899555058, 16.24072330190243185278625729937, 16.91753668168640460051648153884
