| L(s) = 1 | − 18·3-s − 8·4-s + 54·5-s − 28·7-s + 123·9-s − 54·11-s + 144·12-s − 972·15-s + 918·17-s + 30·19-s − 432·20-s + 504·21-s − 486·23-s + 547·25-s − 270·27-s + 224·28-s + 3.24e3·29-s − 546·31-s + 972·33-s − 1.51e3·35-s − 984·36-s − 446·37-s + 2.34e3·43-s + 432·44-s + 6.64e3·45-s + 702·47-s − 4.21e3·49-s + ⋯ |
| L(s) = 1 | − 2·3-s − 1/2·4-s + 2.15·5-s − 4/7·7-s + 1.51·9-s − 0.446·11-s + 12-s − 4.31·15-s + 3.17·17-s + 0.0831·19-s − 1.07·20-s + 8/7·21-s − 0.918·23-s + 0.875·25-s − 0.370·27-s + 2/7·28-s + 3.85·29-s − 0.568·31-s + 0.892·33-s − 1.23·35-s − 0.759·36-s − 0.325·37-s + 1.26·43-s + 0.223·44-s + 3.27·45-s + 0.317·47-s − 1.75·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38416 ^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8738989256\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8738989256\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2^2$ | \( 1 + p^{3} T^{2} + p^{6} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 2 p T + p^{4} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 2 p^{2} T + 67 p T^{2} + 62 p^{3} T^{3} + 1204 p^{2} T^{4} + 62 p^{7} T^{5} + 67 p^{9} T^{6} + 2 p^{14} T^{7} + p^{16} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 54 T + 2369 T^{2} - 75438 T^{3} + 2168484 T^{4} - 75438 p^{4} T^{5} + 2369 p^{8} T^{6} - 54 p^{12} T^{7} + p^{16} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 54 T - 2437 p T^{2} + 23814 T^{3} + 626404692 T^{4} + 23814 p^{4} T^{5} - 2437 p^{9} T^{6} + 54 p^{12} T^{7} + p^{16} T^{8} \) |
| 13 | $D_4\times C_2$ | \( 1 - 51652 T^{2} + 2274555846 T^{4} - 51652 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 54 p T + 493601 T^{2} - 11485422 p T^{3} + 61724271876 T^{4} - 11485422 p^{5} T^{5} + 493601 p^{8} T^{6} - 54 p^{13} T^{7} + p^{16} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 30 T + 66617 T^{2} - 1989510 T^{3} - 12546522252 T^{4} - 1989510 p^{4} T^{5} + 66617 p^{8} T^{6} - 30 p^{12} T^{7} + p^{16} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 486 T + 78265 T^{2} - 195250986 T^{3} - 119466109356 T^{4} - 195250986 p^{4} T^{5} + 78265 p^{8} T^{6} + 486 p^{12} T^{7} + p^{16} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 1620 T + 2066054 T^{2} - 1620 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 546 T + 1193657 T^{2} + 597479610 T^{3} + 436340752596 T^{4} + 597479610 p^{4} T^{5} + 1193657 p^{8} T^{6} + 546 p^{12} T^{7} + p^{16} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 446 T - 3015935 T^{2} - 237928066 T^{3} + 6449986888804 T^{4} - 237928066 p^{4} T^{5} - 3015935 p^{8} T^{6} + 446 p^{12} T^{7} + p^{16} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 9195268 T^{2} + 37043496050310 T^{4} - 9195268 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 1172 T + 3821766 T^{2} - 1172 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 702 T + 7568153 T^{2} - 5197527270 T^{3} + 31807801869972 T^{4} - 5197527270 p^{4} T^{5} + 7568153 p^{8} T^{6} - 702 p^{12} T^{7} + p^{16} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 2754 T - 5299967 T^{2} + 7976903166 T^{3} + 43904732373732 T^{4} + 7976903166 p^{4} T^{5} - 5299967 p^{8} T^{6} - 2754 p^{12} T^{7} + p^{16} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 12366 T + 87438953 T^{2} - 450942278166 T^{3} + 1800614696429652 T^{4} - 450942278166 p^{4} T^{5} + 87438953 p^{8} T^{6} - 12366 p^{12} T^{7} + p^{16} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 126 p T + 39234641 T^{2} - 2462431734 p T^{3} + 462871617507012 T^{4} - 2462431734 p^{5} T^{5} + 39234641 p^{8} T^{6} - 126 p^{13} T^{7} + p^{16} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 5062 T + 1333849 T^{2} - 81053994314 T^{3} - 332413001385740 T^{4} - 81053994314 p^{4} T^{5} + 1333849 p^{8} T^{6} + 5062 p^{12} T^{7} + p^{16} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 9396 T + 71231078 T^{2} - 9396 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 17274 T + 161686097 T^{2} + 1074829823970 T^{3} + 5889761488255716 T^{4} + 1074829823970 p^{4} T^{5} + 161686097 p^{8} T^{6} + 17274 p^{12} T^{7} + p^{16} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 794 T - 69869063 T^{2} + 5876126422 T^{3} + 3428515016079124 T^{4} + 5876126422 p^{4} T^{5} - 69869063 p^{8} T^{6} - 794 p^{12} T^{7} + p^{16} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 153397060 T^{2} + 10369989980918982 T^{4} - 153397060 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 12474 T + 182683793 T^{2} + 1631810023074 T^{3} + 16430717819326692 T^{4} + 1631810023074 p^{4} T^{5} + 182683793 p^{8} T^{6} + 12474 p^{12} T^{7} + p^{16} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 281924740 T^{2} + 35525126061727494 T^{4} - 281924740 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−14.24931730731304178026717873469, −13.91526165140878459477615966428, −13.47264567523626169559698605848, −13.11187538038583761576503023250, −12.67222590269912192973664680883, −12.26812170209811569708195495775, −12.00328351079658417387847940792, −11.73437019802497804966959713482, −11.17680176016329890874710793722, −10.51937692744744366146364942026, −10.07593056200087511253200364713, −9.993828081510207705370444542681, −9.915111920181594314695337083319, −9.331726816700467104543229582288, −8.364604495606938594391378764805, −8.223898496734716742371308947196, −7.34983296192682127925066568101, −6.45202168228227949375328516930, −6.41554897790853692003439734286, −5.58140145221954869577744944150, −5.42076147750614845810174142746, −5.32191428348983662838364376165, −4.00420387118227097261736069547, −2.68841586638820840173826987282, −0.999557796095517861937555837908,
0.999557796095517861937555837908, 2.68841586638820840173826987282, 4.00420387118227097261736069547, 5.32191428348983662838364376165, 5.42076147750614845810174142746, 5.58140145221954869577744944150, 6.41554897790853692003439734286, 6.45202168228227949375328516930, 7.34983296192682127925066568101, 8.223898496734716742371308947196, 8.364604495606938594391378764805, 9.331726816700467104543229582288, 9.915111920181594314695337083319, 9.993828081510207705370444542681, 10.07593056200087511253200364713, 10.51937692744744366146364942026, 11.17680176016329890874710793722, 11.73437019802497804966959713482, 12.00328351079658417387847940792, 12.26812170209811569708195495775, 12.67222590269912192973664680883, 13.11187538038583761576503023250, 13.47264567523626169559698605848, 13.91526165140878459477615966428, 14.24931730731304178026717873469
