| L(s) = 1 | − 4.20e3·5-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 + 1.00e9·49-s + 3.55e8·53-s − 1.94e9·61-s + 4.37e9·65-s + 1.50e9·73-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 + 1.98e10·121-s − 5.26e10·125-s + 127-s + 131-s + 137-s + 139-s + 1.30e11·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1.34·5-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 + 3.55·49-s + 0.851·53-s − 2.30·61-s + 3.76·65-s + 0.725·73-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 + 0.764·121-s − 1.72·125-s + 2.04·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(11-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+5)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.3003560445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3003560445\) |
| \(L(6)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 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
−7.59151253640633797125329200519, −7.42002944327031476116465508100, −7.36098277928006540339221454189, −6.96087497632725882664616223299, −6.77826374218845466781227211748, −6.26112880999946569278887974393, −5.80035071209164936693170105247, −5.51531682198186640084180730477, −5.26426908672570806752066723958, −5.23198059785205187873795353634, −4.87595652875676626907309600166, −4.48864908940318222784005012644, −4.01101540126044032062032727536, −3.76326188180399276995612952811, −3.64990651908492625901474793930, −3.33836166380715164477351080558, −2.76620623188924653910047796374, −2.63010886416183685500157441054, −2.45236941349753818838531013305, −1.82646534777484092490981879290, −1.40824677030437730130253150561, −1.28154460812849413749660133424, −0.867189534644780323097761652410, −0.24521363379071806785904602419, −0.14172703891107289492319971167,
0.14172703891107289492319971167, 0.24521363379071806785904602419, 0.867189534644780323097761652410, 1.28154460812849413749660133424, 1.40824677030437730130253150561, 1.82646534777484092490981879290, 2.45236941349753818838531013305, 2.63010886416183685500157441054, 2.76620623188924653910047796374, 3.33836166380715164477351080558, 3.64990651908492625901474793930, 3.76326188180399276995612952811, 4.01101540126044032062032727536, 4.48864908940318222784005012644, 4.87595652875676626907309600166, 5.23198059785205187873795353634, 5.26426908672570806752066723958, 5.51531682198186640084180730477, 5.80035071209164936693170105247, 6.26112880999946569278887974393, 6.77826374218845466781227211748, 6.96087497632725882664616223299, 7.36098277928006540339221454189, 7.42002944327031476116465508100, 7.59151253640633797125329200519
