| L(s) = 1 | + 108·3-s + 172·7-s + 7.29e3·9-s − 1.00e3·11-s − 1.28e4·13-s + 4.73e3·17-s − 4.11e4·19-s + 1.85e4·21-s + 4.66e4·23-s + 3.93e5·27-s − 1.12e5·29-s + 6.78e4·31-s − 1.08e5·33-s + 1.37e5·37-s − 1.38e6·39-s + 6.28e5·41-s + 1.34e6·43-s − 7.51e5·47-s − 3.67e5·49-s + 5.11e5·51-s − 3.50e5·53-s − 4.44e6·57-s + 1.81e6·59-s − 4.11e5·61-s + 1.25e6·63-s − 1.20e5·67-s + 5.03e6·69-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 0.189·7-s + 10/3·9-s − 0.228·11-s − 1.62·13-s + 0.233·17-s − 1.37·19-s + 0.437·21-s + 0.799·23-s + 3.84·27-s − 0.859·29-s + 0.408·31-s − 0.527·33-s + 0.447·37-s − 3.74·39-s + 1.42·41-s + 2.57·43-s − 1.05·47-s − 0.445·49-s + 0.539·51-s − 0.323·53-s − 3.17·57-s + 1.14·59-s − 0.232·61-s + 0.631·63-s − 0.0490·67-s + 1.84·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(16.37643510\) |
| \(L(\frac12)\) |
\(\approx\) |
\(16.37643510\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2 \wr S_4$ | \( 1 - 172 T + 396810 T^{2} - 309524792 T^{3} - 209164259453 T^{4} - 309524792 p^{7} T^{5} + 396810 p^{14} T^{6} - 172 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 1008 T + 43126532 T^{2} - 45253585616 T^{3} + 875739654641430 T^{4} - 45253585616 p^{7} T^{5} + 43126532 p^{14} T^{6} + 1008 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 12836 T + 112714970 T^{2} + 753978899136 T^{3} + 525672927715295 p T^{4} + 753978899136 p^{7} T^{5} + 112714970 p^{14} T^{6} + 12836 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 4736 T + 26366812 T^{2} - 5608802064640 T^{3} + 238208497523355398 T^{4} - 5608802064640 p^{7} T^{5} + 26366812 p^{14} T^{6} - 4736 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 41140 T + 2613821242 T^{2} + 82349248695224 T^{3} + 3532189069041985091 T^{4} + 82349248695224 p^{7} T^{5} + 2613821242 p^{14} T^{6} + 41140 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 46656 T + 9671048180 T^{2} - 258454440332224 T^{3} + 40872887850723542022 T^{4} - 258454440332224 p^{7} T^{5} + 9671048180 p^{14} T^{6} - 46656 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 112912 T + 57647705740 T^{2} + 5869822421936432 T^{3} + \)\(13\!\cdots\!14\)\( T^{4} + 5869822421936432 p^{7} T^{5} + 57647705740 p^{14} T^{6} + 112912 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 67812 T + 84989040634 T^{2} - 5352885423949096 T^{3} + \)\(32\!\cdots\!31\)\( T^{4} - 5352885423949096 p^{7} T^{5} + 84989040634 p^{14} T^{6} - 67812 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 137864 T + 346403984812 T^{2} - 39000456799221336 T^{3} + \)\(47\!\cdots\!14\)\( T^{4} - 39000456799221336 p^{7} T^{5} + 346403984812 p^{14} T^{6} - 137864 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 628760 T + 146116200020 T^{2} + 120665676982917624 T^{3} - \)\(75\!\cdots\!10\)\( T^{4} + 120665676982917624 p^{7} T^{5} + 146116200020 p^{14} T^{6} - 628760 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 1341932 T + 1069809013818 T^{2} - 622096830646297960 T^{3} + \)\(30\!\cdots\!83\)\( T^{4} - 622096830646297960 p^{7} T^{5} + 1069809013818 p^{14} T^{6} - 1341932 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 751576 T + 8869152556 p T^{2} + 575980853001016664 T^{3} + \)\(59\!\cdots\!94\)\( T^{4} + 575980853001016664 p^{7} T^{5} + 8869152556 p^{15} T^{6} + 751576 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 350888 T + 50648843476 p T^{2} + 1240856260712300568 T^{3} + \)\(42\!\cdots\!34\)\( T^{4} + 1240856260712300568 p^{7} T^{5} + 50648843476 p^{15} T^{6} + 350888 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 1810376 T + 8125932242148 T^{2} - 11530949651991542952 T^{3} + \)\(28\!\cdots\!50\)\( T^{4} - 11530949651991542952 p^{7} T^{5} + 8125932242148 p^{14} T^{6} - 1810376 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 + 411860 T + 4054047314730 T^{2} + 7283543416535544544 T^{3} + \)\(53\!\cdots\!95\)\( T^{4} + 7283543416535544544 p^{7} T^{5} + 4054047314730 p^{14} T^{6} + 411860 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 120820 T + 13607642614906 T^{2} - 3535073645239426312 T^{3} + \)\(92\!\cdots\!75\)\( T^{4} - 3535073645239426312 p^{7} T^{5} + 13607642614906 p^{14} T^{6} + 120820 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 107944 T + 31844848180556 T^{2} + 2714803125792069192 T^{3} + \)\(41\!\cdots\!38\)\( T^{4} + 2714803125792069192 p^{7} T^{5} + 31844848180556 p^{14} T^{6} + 107944 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 470792 T + 35574415090012 T^{2} - 25765042327396072440 T^{3} + \)\(53\!\cdots\!26\)\( T^{4} - 25765042327396072440 p^{7} T^{5} + 35574415090012 p^{14} T^{6} - 470792 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 8235872 T + 95921708136380 T^{2} - \)\(47\!\cdots\!72\)\( T^{3} + \)\(29\!\cdots\!54\)\( T^{4} - \)\(47\!\cdots\!72\)\( p^{7} T^{5} + 95921708136380 p^{14} T^{6} - 8235872 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 5735800 T + 75850061342724 T^{2} - \)\(19\!\cdots\!04\)\( T^{3} + \)\(21\!\cdots\!50\)\( T^{4} - \)\(19\!\cdots\!04\)\( p^{7} T^{5} + 75850061342724 p^{14} T^{6} - 5735800 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 12421472 T + 56583490013604 T^{2} + \)\(39\!\cdots\!32\)\( T^{3} - \)\(48\!\cdots\!90\)\( T^{4} + \)\(39\!\cdots\!32\)\( p^{7} T^{5} + 56583490013604 p^{14} T^{6} - 12421472 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + 2769732 T + 78831585919562 T^{2} - 89896504269839223024 T^{3} + \)\(97\!\cdots\!23\)\( T^{4} - 89896504269839223024 p^{7} T^{5} + 78831585919562 p^{14} T^{6} + 2769732 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.90056748588598593648490343647, −6.10802871102052922290699535222, −6.07179053914995430565913152278, −5.99877358259266723157015082226, −5.73332620661480943263791180925, −4.90533645874331597349622506589, −4.87075144647122485385187301848, −4.84436776536498424297992974180, −4.83158523323025085892787696250, −4.18050505839040028926979210775, −3.91600938545916151815206187965, −3.82308163665245772065940614145, −3.66199113411064311526771989094, −3.15521481518496685537078538393, −2.90211936294401884767621478239, −2.76905135962191831234093349111, −2.54516119659293581862186849830, −2.17004915979545580550889795587, −2.04639214561241573221624205146, −1.78864268759221472125511407833, −1.68403346865417341255036990664, −0.832129863354923922459144232363, −0.803864967783139114256139431859, −0.74328687353119055443950259186, −0.23379389947935664224491832146,
0.23379389947935664224491832146, 0.74328687353119055443950259186, 0.803864967783139114256139431859, 0.832129863354923922459144232363, 1.68403346865417341255036990664, 1.78864268759221472125511407833, 2.04639214561241573221624205146, 2.17004915979545580550889795587, 2.54516119659293581862186849830, 2.76905135962191831234093349111, 2.90211936294401884767621478239, 3.15521481518496685537078538393, 3.66199113411064311526771989094, 3.82308163665245772065940614145, 3.91600938545916151815206187965, 4.18050505839040028926979210775, 4.83158523323025085892787696250, 4.84436776536498424297992974180, 4.87075144647122485385187301848, 4.90533645874331597349622506589, 5.73332620661480943263791180925, 5.99877358259266723157015082226, 6.07179053914995430565913152278, 6.10802871102052922290699535222, 6.90056748588598593648490343647
