| L(s) = 1 | + 32·2-s + 640·4-s − 100·5-s − 1.37e3·7-s + 1.02e4·8-s − 3.20e3·10-s − 6.44e3·11-s + 6.59e3·13-s − 4.39e4·14-s + 1.43e5·16-s − 2.21e4·17-s − 1.57e4·19-s − 6.40e4·20-s − 2.06e5·22-s + 1.07e5·23-s − 1.38e5·25-s + 2.11e5·26-s − 8.78e5·28-s + 1.63e5·29-s + 3.54e4·31-s + 1.83e6·32-s − 7.09e5·34-s + 1.37e5·35-s + 3.71e5·37-s − 5.03e5·38-s − 1.02e6·40-s − 7.92e4·41-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 5·4-s − 0.357·5-s − 1.51·7-s + 7.07·8-s − 1.01·10-s − 1.46·11-s + 0.832·13-s − 4.27·14-s + 35/4·16-s − 1.09·17-s − 0.526·19-s − 1.78·20-s − 4.13·22-s + 1.83·23-s − 1.77·25-s + 2.35·26-s − 7.55·28-s + 1.24·29-s + 0.213·31-s + 9.89·32-s − 3.09·34-s + 0.540·35-s + 1.20·37-s − 1.48·38-s − 2.52·40-s − 0.179·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{12} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(48.29410225\) |
| \(L(\frac12)\) |
\(\approx\) |
\(48.29410225\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 4 p^{2} T + 29671 p T^{2} + 732784 p^{2} T^{3} + 3749612 p^{5} T^{4} + 732784 p^{9} T^{5} + 29671 p^{15} T^{6} + 4 p^{23} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 586 p T + 37769756 T^{2} + 228514659326 T^{3} + 1078167507581590 T^{4} + 228514659326 p^{7} T^{5} + 37769756 p^{14} T^{6} + 586 p^{22} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 6594 T + 19891447 T^{2} - 364106053224 T^{3} + 5942923970112420 T^{4} - 364106053224 p^{7} T^{5} + 19891447 p^{14} T^{6} - 6594 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 22184 T + 974685890 T^{2} + 17362692213776 T^{3} + 550385287420035907 T^{4} + 17362692213776 p^{7} T^{5} + 974685890 p^{14} T^{6} + 22184 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 15730 T + 1980440200 T^{2} + 11316965148682 T^{3} + 1806787307706254590 T^{4} + 11316965148682 p^{7} T^{5} + 1980440200 p^{14} T^{6} + 15730 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 107176 T + 7550967635 T^{2} - 263716086104380 T^{3} + 15096085918285347364 T^{4} - 263716086104380 p^{7} T^{5} + 7550967635 p^{14} T^{6} - 107176 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 163844 T + 54237375941 T^{2} - 7533654826868990 T^{3} + \)\(13\!\cdots\!96\)\( T^{4} - 7533654826868990 p^{7} T^{5} + 54237375941 p^{14} T^{6} - 163844 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 35424 T + 78920739349 T^{2} - 2293037033326794 T^{3} + \)\(28\!\cdots\!76\)\( T^{4} - 2293037033326794 p^{7} T^{5} + 78920739349 p^{14} T^{6} - 35424 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 371678 T + 36654646105 T^{2} + 4638931743405058 T^{3} - 27536034361369275620 T^{4} + 4638931743405058 p^{7} T^{5} + 36654646105 p^{14} T^{6} - 371678 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 1932 p T + 601991836139 T^{2} + 27938884514442804 T^{3} + \)\(16\!\cdots\!80\)\( T^{4} + 27938884514442804 p^{7} T^{5} + 601991836139 p^{14} T^{6} + 1932 p^{22} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1186652 T + 1017715522390 T^{2} - 641363373474832928 T^{3} + \)\(35\!\cdots\!11\)\( T^{4} - 641363373474832928 p^{7} T^{5} + 1017715522390 p^{14} T^{6} - 1186652 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 952110 T + 1004903355581 T^{2} - 895872494580890022 T^{3} + \)\(78\!\cdots\!80\)\( T^{4} - 895872494580890022 p^{7} T^{5} + 1004903355581 p^{14} T^{6} - 952110 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 155082 T + 3887296735253 T^{2} - 501972673899779730 T^{3} + \)\(64\!\cdots\!88\)\( T^{4} - 501972673899779730 p^{7} T^{5} + 3887296735253 p^{14} T^{6} - 155082 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1193848 T + 9190200912434 T^{2} + 7545670251918781024 T^{3} + \)\(32\!\cdots\!07\)\( T^{4} + 7545670251918781024 p^{7} T^{5} + 9190200912434 p^{14} T^{6} + 1193848 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 129102 T + 3936176501464 T^{2} + 69943666393612350 p T^{3} + \)\(89\!\cdots\!86\)\( T^{4} + 69943666393612350 p^{8} T^{5} + 3936176501464 p^{14} T^{6} - 129102 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 2555300 T + 10181443348363 T^{2} + 19491175790322912344 T^{3} + \)\(44\!\cdots\!52\)\( T^{4} + 19491175790322912344 p^{7} T^{5} + 10181443348363 p^{14} T^{6} + 2555300 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2975872 T + 37134034488479 T^{2} - 79994336615079452740 T^{3} + \)\(50\!\cdots\!76\)\( T^{4} - 79994336615079452740 p^{7} T^{5} + 37134034488479 p^{14} T^{6} - 2975872 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 6428020 T + 50916284945092 T^{2} - \)\(18\!\cdots\!60\)\( T^{3} + \)\(83\!\cdots\!30\)\( T^{4} - \)\(18\!\cdots\!60\)\( p^{7} T^{5} + 50916284945092 p^{14} T^{6} - 6428020 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 6843240 T + 65675461015531 T^{2} - \)\(31\!\cdots\!80\)\( T^{3} + \)\(18\!\cdots\!96\)\( T^{4} - \)\(31\!\cdots\!80\)\( p^{7} T^{5} + 65675461015531 p^{14} T^{6} - 6843240 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 11642074 T + 59294184135017 T^{2} - 2971274369590018450 T^{3} - \)\(81\!\cdots\!40\)\( T^{4} - 2971274369590018450 p^{7} T^{5} + 59294184135017 p^{14} T^{6} - 11642074 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 21148548 T + 217796908793279 T^{2} - \)\(15\!\cdots\!32\)\( T^{3} + \)\(98\!\cdots\!40\)\( T^{4} - \)\(15\!\cdots\!32\)\( p^{7} T^{5} + 217796908793279 p^{14} T^{6} - 21148548 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 5112362 T + 272857671132172 T^{2} + \)\(11\!\cdots\!66\)\( T^{3} + \)\(31\!\cdots\!98\)\( T^{4} + \)\(11\!\cdots\!66\)\( p^{7} T^{5} + 272857671132172 p^{14} T^{6} + 5112362 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.77604525853188241827741588131, −6.49567130761119733480325640958, −6.40126198665058806313381934131, −6.29935235283886523838810634156, −5.99120967991115494030008398685, −5.53904171798734858658736451156, −5.45983886513135851037210149026, −5.36558931264434742668465764862, −4.94681935910197363926215222052, −4.44388633268838304216968572259, −4.31752661842578960896654583945, −4.23850031830547121583346524559, −4.18357316403807972985803284166, −3.28736235789987481276694668005, −3.27268505241111926766672672241, −3.22175692185517460789574763687, −3.16226595643089455648031981615, −2.35963145412013521695055585640, −2.22615927869445392953529992026, −2.11336436870991825355468994730, −2.04780677743601341068179679897, −1.02962583352068960197067786283, −0.841236626006356053782253646855, −0.68043247327875712588227256680, −0.35505705168990815222934679376,
0.35505705168990815222934679376, 0.68043247327875712588227256680, 0.841236626006356053782253646855, 1.02962583352068960197067786283, 2.04780677743601341068179679897, 2.11336436870991825355468994730, 2.22615927869445392953529992026, 2.35963145412013521695055585640, 3.16226595643089455648031981615, 3.22175692185517460789574763687, 3.27268505241111926766672672241, 3.28736235789987481276694668005, 4.18357316403807972985803284166, 4.23850031830547121583346524559, 4.31752661842578960896654583945, 4.44388633268838304216968572259, 4.94681935910197363926215222052, 5.36558931264434742668465764862, 5.45983886513135851037210149026, 5.53904171798734858658736451156, 5.99120967991115494030008398685, 6.29935235283886523838810634156, 6.40126198665058806313381934131, 6.49567130761119733480325640958, 6.77604525853188241827741588131
