| L(s) = 1 | − 28·3-s − 42·5-s + 112·7-s + 573·9-s − 660·11-s − 1.28e3·13-s + 1.17e3·15-s + 210·17-s − 3.72e3·19-s − 3.13e3·21-s − 24·23-s + 2.76e3·25-s − 1.25e4·27-s + 1.10e4·29-s − 2.80e3·31-s + 1.84e4·33-s − 4.70e3·35-s + 1.32e4·37-s + 3.60e4·39-s + 8.23e3·41-s + 2.68e4·43-s − 2.40e4·45-s − 8.06e3·47-s − 2.42e4·49-s − 5.88e3·51-s + 5.39e4·53-s + 2.77e4·55-s + ⋯ |
| L(s) = 1 | − 1.79·3-s − 0.751·5-s + 0.863·7-s + 2.35·9-s − 1.64·11-s − 2.11·13-s + 1.34·15-s + 0.176·17-s − 2.36·19-s − 1.55·21-s − 0.00946·23-s + 0.885·25-s − 3.31·27-s + 2.44·29-s − 0.523·31-s + 2.95·33-s − 0.649·35-s + 1.58·37-s + 3.79·39-s + 0.764·41-s + 2.21·43-s − 1.77·45-s − 0.532·47-s − 1.44·49-s − 0.316·51-s + 2.63·53-s + 1.23·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 614656 ^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.06336617373\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06336617373\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - 8 p T + p^{5} T^{2} )^{2} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 28 T + 211 T^{2} + 812 p T^{3} + 8752 p^{2} T^{4} + 812 p^{6} T^{5} + 211 p^{10} T^{6} + 28 p^{15} T^{7} + p^{20} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 + 42 T - 1003 T^{2} - 146286 T^{3} - 7621836 T^{4} - 146286 p^{5} T^{5} - 1003 p^{10} T^{6} + 42 p^{15} T^{7} + p^{20} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 60 p T + 52667 T^{2} + 3649860 p T^{3} + 50546442288 T^{4} + 3649860 p^{6} T^{5} + 52667 p^{10} T^{6} + 60 p^{16} T^{7} + p^{20} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 644 T + 595134 T^{2} + 644 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 210 T - 2743855 T^{2} + 10869390 T^{3} + 5694083487876 T^{4} + 10869390 p^{5} T^{5} - 2743855 p^{10} T^{6} - 210 p^{15} T^{7} + p^{20} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 196 p T + 5457763 T^{2} + 677810140 p T^{3} + 30265003847344 T^{4} + 677810140 p^{6} T^{5} + 5457763 p^{10} T^{6} + 196 p^{16} T^{7} + p^{20} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 24 T - 10516873 T^{2} - 56525688 T^{3} + 69186514920384 T^{4} - 56525688 p^{5} T^{5} - 10516873 p^{10} T^{6} + 24 p^{15} T^{7} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 5532 T + 36367390 T^{2} - 5532 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 2800 T - 30179873 T^{2} - 53867601200 T^{3} + 401429912115328 T^{4} - 53867601200 p^{5} T^{5} - 30179873 p^{10} T^{6} + 2800 p^{15} T^{7} + p^{20} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 13238 T + 2167093 T^{2} - 455250014606 T^{3} + 12995097229723444 T^{4} - 455250014606 p^{5} T^{5} + 2167093 p^{10} T^{6} - 13238 p^{15} T^{7} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 4116 T + 163369462 T^{2} - 4116 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6716 T + p^{5} T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 + 8064 T + 128729519 T^{2} - 4212564547968 T^{3} - 63455130075921792 T^{4} - 4212564547968 p^{5} T^{5} + 128729519 p^{10} T^{6} + 8064 p^{15} T^{7} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 53958 T + 1379702981 T^{2} - 37520871422526 T^{3} + 953847892489790388 T^{4} - 37520871422526 p^{5} T^{5} + 1379702981 p^{10} T^{6} - 53958 p^{15} T^{7} + p^{20} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 36036 T - 113258965 T^{2} - 648516000132 T^{3} + 553814151761271216 T^{4} - 648516000132 p^{5} T^{5} - 113258965 p^{10} T^{6} + 36036 p^{15} T^{7} + p^{20} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 + 83986 T + 3603119341 T^{2} + 147927586544458 T^{3} + 5235321292627805428 T^{4} + 147927586544458 p^{5} T^{5} + 3603119341 p^{10} T^{6} + 83986 p^{15} T^{7} + p^{20} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 2660 T - 2654517485 T^{2} + 102827963140 T^{3} + 5251687000360966576 T^{4} + 102827963140 p^{5} T^{5} - 2654517485 p^{10} T^{6} - 2660 p^{15} T^{7} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 1008 p T + 3116937742 T^{2} - 1008 p^{6} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 + 31318 T - 16662743 T^{2} - 98609837824442 T^{3} - 5335963862325578492 T^{4} - 98609837824442 p^{5} T^{5} - 16662743 p^{10} T^{6} + 31318 p^{15} T^{7} + p^{20} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 + 51136 T + 637460815 T^{2} - 213578867870912 T^{3} - 13604213472728169344 T^{4} - 213578867870912 p^{5} T^{5} + 637460815 p^{10} T^{6} + 51136 p^{15} T^{7} + p^{20} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 6216 T + 6437179414 T^{2} - 6216 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 166278 T + 9871423481 T^{2} + 1098903185221590 T^{3} + \)\(13\!\cdots\!92\)\( T^{4} + 1098903185221590 p^{5} T^{5} + 9871423481 p^{10} T^{6} + 166278 p^{15} T^{7} + p^{20} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 8260 T + 11911603014 T^{2} - 8260 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
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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
−11.76425798798792929836030098272, −11.72317954691420239837339796002, −11.17462790716574315087503796615, −10.72835068401309636092867991536, −10.69025674196006612850213690221, −10.46309772491494831805346348245, −10.03804765478070224401771948908, −9.377911125421513572844738929554, −9.375985378834124271067946470231, −8.433881689701672670447128717991, −8.141526752538879996028832892595, −7.74554255132664598056058465262, −7.55618606425390359576298185921, −7.01128312814749977758585862235, −6.63770532675173131210176822447, −6.06975733832410476113673053991, −5.72545035227542400301365297762, −4.94184237433482742391751810834, −4.91130594354686252244896397626, −4.41451122090685004811587990090, −4.09354277326655723022911140888, −2.60291890222245140233199132931, −2.42721347099846571737991193663, −1.13347534732436320568214951797, −0.11318291294161122895451000317,
0.11318291294161122895451000317, 1.13347534732436320568214951797, 2.42721347099846571737991193663, 2.60291890222245140233199132931, 4.09354277326655723022911140888, 4.41451122090685004811587990090, 4.91130594354686252244896397626, 4.94184237433482742391751810834, 5.72545035227542400301365297762, 6.06975733832410476113673053991, 6.63770532675173131210176822447, 7.01128312814749977758585862235, 7.55618606425390359576298185921, 7.74554255132664598056058465262, 8.141526752538879996028832892595, 8.433881689701672670447128717991, 9.375985378834124271067946470231, 9.377911125421513572844738929554, 10.03804765478070224401771948908, 10.46309772491494831805346348245, 10.69025674196006612850213690221, 10.72835068401309636092867991536, 11.17462790716574315087503796615, 11.72317954691420239837339796002, 11.76425798798792929836030098272
