| L(s) = 1 | + 4.20e3·5-s + 2.34e4·9-s − 1.04e6·13-s − 4.39e6·17-s + 9.19e6·25-s + 3.11e7·29-s − 1.66e8·37-s + 1.72e8·41-s + 9.85e7·45-s + 1.00e9·49-s − 3.55e8·53-s − 1.94e9·61-s − 4.37e9·65-s + 1.50e9·73-s − 2.91e9·81-s − 1.84e10·85-s + 6.04e9·89-s − 8.29e9·97-s + 5.62e10·101-s − 5.48e9·109-s − 3.60e10·113-s − 2.44e10·117-s + 1.98e10·121-s + 5.26e10·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.397·9-s − 2.80·13-s − 3.09·17-s + 0.941·25-s + 1.51·29-s − 2.39·37-s + 1.49·41-s + 0.533·45-s + 3.55·49-s − 0.851·53-s − 2.30·61-s − 3.76·65-s + 0.725·73-s − 0.835·81-s − 4.15·85-s + 1.08·89-s − 0.965·97-s + 5.35·101-s − 0.356·109-s − 1.95·113-s − 1.11·117-s + 0.764·121-s + 1.72·125-s + 2.04·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 65536 ^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.7338721252\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7338721252\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| good | 3 | $D_4\times C_2$ | \( 1 - 7820 p T^{2} + 42745222 p^{4} T^{4} - 7820 p^{21} T^{6} + p^{40} T^{8} \) |
| 5 | $D_{4}$ | \( ( 1 - 84 p^{2} T + 403486 p T^{2} - 84 p^{12} T^{3} + p^{20} T^{4} )^{2} \) |
| 7 | $D_4\times C_2$ | \( 1 - 1005064132 T^{2} + 8393882548202022 p^{2} T^{4} - 1005064132 p^{20} T^{6} + p^{40} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 19840803940 T^{2} + \)\(44\!\cdots\!82\)\( T^{4} - 19840803940 p^{20} T^{6} + p^{40} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 520492 T + 254812664694 T^{2} + 520492 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 + 2195004 T + 5145278472902 T^{2} + 2195004 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 3663898849060 T^{2} + \)\(12\!\cdots\!02\)\( p^{2} T^{4} - 3663898849060 p^{20} T^{6} + p^{40} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 82119401461060 T^{2} + \)\(46\!\cdots\!82\)\( T^{4} - 82119401461060 p^{20} T^{6} + p^{40} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 15568404 T + 902007355177526 T^{2} - 15568404 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1229395698322180 T^{2} + \)\(92\!\cdots\!02\)\( T^{4} - 1229395698322180 p^{20} T^{6} + p^{40} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 83083340 T + 6979856299974678 T^{2} + 83083340 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 86387364 T + 18423149137257446 T^{2} - 86387364 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 + 888082514945180 T^{2} - \)\(50\!\cdots\!78\)\( T^{4} + 888082514945180 p^{20} T^{6} + p^{40} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 196091355038080132 T^{2} + \)\(15\!\cdots\!38\)\( T^{4} - 196091355038080132 p^{20} T^{6} + p^{40} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 177944076 T + 213352476870696662 T^{2} + 177944076 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 815549572994663140 T^{2} + \)\(68\!\cdots\!82\)\( T^{4} - 815549572994663140 p^{20} T^{6} + p^{40} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 974574188 T + 1661874172242478518 T^{2} + 974574188 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 5292742361877207460 T^{2} + \)\(13\!\cdots\!82\)\( T^{4} - 5292742361877207460 p^{20} T^{6} + p^{40} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 11040185613269882308 T^{2} + \)\(50\!\cdots\!18\)\( T^{4} - 11040185613269882308 p^{20} T^{6} + p^{40} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 - 752509604 T + 7133539166909268582 T^{2} - 752509604 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 79 | $D_4\times C_2$ | \( 1 + 5042168236353513596 T^{2} - \)\(20\!\cdots\!34\)\( p^{2} T^{4} + 5042168236353513596 p^{20} T^{6} + p^{40} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 45281007302789483812 T^{2} + \)\(92\!\cdots\!58\)\( T^{4} - 45281007302789483812 p^{20} T^{6} + p^{40} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 - 3020717028 T + 27059817309971120678 T^{2} - 3020717028 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 4146123772 T + \)\(13\!\cdots\!94\)\( T^{2} + 4146123772 p^{10} T^{3} + p^{20} T^{4} )^{2} \) |
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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
−12.16660310188118915971574994012, −11.76308808771619221251441585596, −11.07874599739570773075978228905, −10.60953430757752636473749483221, −10.58286418060353812232858296586, −9.998594944504876633032118969232, −9.755150154559795966936835243180, −9.147324399292433709100372209070, −8.986285561238946826011777779989, −8.700346315919732513757505959186, −7.927923402745418275615064704655, −7.28773091429851534283290942161, −6.93794030710311867454845399939, −6.89964890220859404897507991748, −5.98059704988160615002328850388, −5.84958137220950197859296000347, −4.85806993576184713190034775078, −4.68758227200634109429992512006, −4.48437215658643818055072984822, −3.40870268209642714033529020828, −2.45967805455503693161608709367, −2.31777551149037361156197018100, −2.03607356497616562148237030664, −1.04463879815843202073031552507, −0.18548361289326149450554229332,
0.18548361289326149450554229332, 1.04463879815843202073031552507, 2.03607356497616562148237030664, 2.31777551149037361156197018100, 2.45967805455503693161608709367, 3.40870268209642714033529020828, 4.48437215658643818055072984822, 4.68758227200634109429992512006, 4.85806993576184713190034775078, 5.84958137220950197859296000347, 5.98059704988160615002328850388, 6.89964890220859404897507991748, 6.93794030710311867454845399939, 7.28773091429851534283290942161, 7.927923402745418275615064704655, 8.700346315919732513757505959186, 8.986285561238946826011777779989, 9.147324399292433709100372209070, 9.755150154559795966936835243180, 9.998594944504876633032118969232, 10.58286418060353812232858296586, 10.60953430757752636473749483221, 11.07874599739570773075978228905, 11.76308808771619221251441585596, 12.16660310188118915971574994012
