| L(s) = 1 | + 188·5-s + 180·7-s − 1.46e3·11-s − 440·13-s + 9.92e3·17-s + 2.03e3·19-s − 2.42e4·23-s − 1.26e5·25-s + 5.80e4·29-s − 6.39e4·31-s + 3.38e4·35-s + 3.51e5·37-s − 9.84e3·41-s − 1.43e5·43-s + 3.76e5·47-s − 7.77e5·49-s − 4.98e5·53-s − 2.74e5·55-s + 4.39e5·59-s + 2.95e6·61-s − 8.27e4·65-s − 7.47e6·67-s + 1.14e5·71-s + 8.32e6·73-s − 2.62e5·77-s − 1.29e7·79-s + 4.04e5·83-s + ⋯ |
| L(s) = 1 | + 0.672·5-s + 0.198·7-s − 0.330·11-s − 0.0555·13-s + 0.490·17-s + 0.0679·19-s − 0.415·23-s − 1.61·25-s + 0.442·29-s − 0.385·31-s + 0.133·35-s + 1.14·37-s − 0.0222·41-s − 0.275·43-s + 0.528·47-s − 0.944·49-s − 0.460·53-s − 0.222·55-s + 0.278·59-s + 1.66·61-s − 0.0373·65-s − 3.03·67-s + 0.0379·71-s + 2.50·73-s − 0.0656·77-s − 2.95·79-s + 0.0776·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{12}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.927524502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.927524502\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 - 188 T + 161774 T^{2} - 5049736 p T^{3} + 708841399 p^{2} T^{4} - 5049736 p^{8} T^{5} + 161774 p^{14} T^{6} - 188 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 180 T + 810190 T^{2} + 303283440 T^{3} + 286871165871 T^{4} + 303283440 p^{7} T^{5} + 810190 p^{14} T^{6} - 180 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 1460 T + 17383082 T^{2} + 19564246520 T^{3} + 397120723371763 T^{4} + 19564246520 p^{7} T^{5} + 17383082 p^{14} T^{6} + 1460 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 440 T + 123007252 T^{2} + 340873142600 T^{3} + 8942999915138326 T^{4} + 340873142600 p^{7} T^{5} + 123007252 p^{14} T^{6} + 440 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 584 p T + 864377732 T^{2} - 9883213484248 T^{3} + 428235012760844998 T^{4} - 9883213484248 p^{7} T^{5} + 864377732 p^{14} T^{6} - 584 p^{22} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 2032 T + 492348484 T^{2} - 19482265320496 T^{3} + 539543634776263606 T^{4} - 19482265320496 p^{7} T^{5} + 492348484 p^{14} T^{6} - 2032 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 24224 T + 6094785860 T^{2} + 5525452855520 T^{3} + 22461607737886090918 T^{4} + 5525452855520 p^{7} T^{5} + 6094785860 p^{14} T^{6} + 24224 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 58080 T + 46574936972 T^{2} - 2367862042983840 T^{3} + \)\(11\!\cdots\!10\)\( T^{4} - 2367862042983840 p^{7} T^{5} + 46574936972 p^{14} T^{6} - 58080 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 63932 T + 70293178678 T^{2} + 4029053281758704 T^{3} + \)\(26\!\cdots\!95\)\( T^{4} + 4029053281758704 p^{7} T^{5} + 70293178678 p^{14} T^{6} + 63932 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 351600 T + 256539890236 T^{2} - 57226593360030480 T^{3} + \)\(30\!\cdots\!54\)\( T^{4} - 57226593360030480 p^{7} T^{5} + 256539890236 p^{14} T^{6} - 351600 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 240 p T + 316928115452 T^{2} + 134189828488216080 T^{3} + \)\(41\!\cdots\!98\)\( T^{4} + 134189828488216080 p^{7} T^{5} + 316928115452 p^{14} T^{6} + 240 p^{22} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 + 143800 T + 267325723924 T^{2} + 59704011244706200 T^{3} + \)\(16\!\cdots\!42\)\( T^{4} + 59704011244706200 p^{7} T^{5} + 267325723924 p^{14} T^{6} + 143800 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 376056 T + 1368484493012 T^{2} - 149893241396174712 T^{3} + \)\(81\!\cdots\!18\)\( T^{4} - 149893241396174712 p^{7} T^{5} + 1368484493012 p^{14} T^{6} - 376056 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 498916 T + 1372686598838 T^{2} + 455430190165180984 T^{3} + \)\(25\!\cdots\!43\)\( T^{4} + 455430190165180984 p^{7} T^{5} + 1372686598838 p^{14} T^{6} + 498916 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 439360 T + 4031299551212 T^{2} - 5591600751915841600 T^{3} + \)\(99\!\cdots\!58\)\( T^{4} - 5591600751915841600 p^{7} T^{5} + 4031299551212 p^{14} T^{6} - 439360 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 2953936 T + 11575683909844 T^{2} - 20242293775381965616 T^{3} + \)\(49\!\cdots\!90\)\( T^{4} - 20242293775381965616 p^{7} T^{5} + 11575683909844 p^{14} T^{6} - 2953936 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 + 7470200 T + 28473879727156 T^{2} + 67915623690639464600 T^{3} + \)\(15\!\cdots\!14\)\( T^{4} + 67915623690639464600 p^{7} T^{5} + 28473879727156 p^{14} T^{6} + 7470200 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 114440 T + 22926671765660 T^{2} - 19822336073470036520 T^{3} + \)\(25\!\cdots\!34\)\( T^{4} - 19822336073470036520 p^{7} T^{5} + 22926671765660 p^{14} T^{6} - 114440 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 8322700 T + 46541221526530 T^{2} - \)\(15\!\cdots\!00\)\( T^{3} + \)\(54\!\cdots\!43\)\( T^{4} - \)\(15\!\cdots\!00\)\( p^{7} T^{5} + 46541221526530 p^{14} T^{6} - 8322700 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 12953560 T + 93975751679740 T^{2} + \)\(47\!\cdots\!40\)\( T^{3} + \)\(20\!\cdots\!62\)\( T^{4} + \)\(47\!\cdots\!40\)\( p^{7} T^{5} + 93975751679740 p^{14} T^{6} + 12953560 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 404444 T + 79360104911738 T^{2} + 35911034543000660824 T^{3} + \)\(28\!\cdots\!23\)\( T^{4} + 35911034543000660824 p^{7} T^{5} + 79360104911738 p^{14} T^{6} - 404444 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 1770360 T + 126405733766540 T^{2} - 3344563390537508040 T^{3} + \)\(69\!\cdots\!82\)\( T^{4} - 3344563390537508040 p^{7} T^{5} + 126405733766540 p^{14} T^{6} - 1770360 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 25436060 T + 294355596161194 T^{2} - \)\(13\!\cdots\!60\)\( T^{3} + \)\(45\!\cdots\!95\)\( T^{4} - \)\(13\!\cdots\!60\)\( p^{7} T^{5} + 294355596161194 p^{14} T^{6} - 25436060 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−6.99268385262557860460380411295, −6.47181148213308755443765544033, −6.30193708172484952262112148815, −6.08449517754973515136678158832, −5.92732932120051396432312931872, −5.65548914416295856139781378348, −5.42123540425642617230772075807, −5.16107257759087848085379740404, −4.86941667026922781481416457638, −4.56945023935734221387139553143, −4.36742596740254507328263914782, −4.01459145386444179583772715723, −3.83380330605838762477685687779, −3.52897635858437264951492974250, −3.01074790025104509605914233641, −2.88922319481627530925639697148, −2.84845248284274047787699126479, −2.10466577097064446687891469704, −2.01645644087493436250356279615, −1.74300908046406242000423559731, −1.66508681108437219819586896213, −1.11543899910247037277071991176, −0.58682018048530566572085593626, −0.56406711082965792437054763021, −0.38404592761616521944947190936,
0.38404592761616521944947190936, 0.56406711082965792437054763021, 0.58682018048530566572085593626, 1.11543899910247037277071991176, 1.66508681108437219819586896213, 1.74300908046406242000423559731, 2.01645644087493436250356279615, 2.10466577097064446687891469704, 2.84845248284274047787699126479, 2.88922319481627530925639697148, 3.01074790025104509605914233641, 3.52897635858437264951492974250, 3.83380330605838762477685687779, 4.01459145386444179583772715723, 4.36742596740254507328263914782, 4.56945023935734221387139553143, 4.86941667026922781481416457638, 5.16107257759087848085379740404, 5.42123540425642617230772075807, 5.65548914416295856139781378348, 5.92732932120051396432312931872, 6.08449517754973515136678158832, 6.30193708172484952262112148815, 6.47181148213308755443765544033, 6.99268385262557860460380411295
