| L(s) = 1 | − 32·2-s + 108·3-s + 640·4-s − 162·5-s − 3.45e3·6-s + 1.21e3·7-s − 1.02e4·8-s + 7.29e3·9-s + 5.18e3·10-s + 1.82e3·11-s + 6.91e4·12-s − 6.50e3·13-s − 3.89e4·14-s − 1.74e4·15-s + 1.43e5·16-s + 1.28e4·17-s − 2.33e5·18-s + 7.44e4·19-s − 1.03e5·20-s + 1.31e5·21-s − 5.82e4·22-s + 4.86e4·23-s − 1.10e6·24-s − 4.68e4·25-s + 2.08e5·26-s + 3.93e5·27-s + 7.79e5·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 2.30·3-s + 5·4-s − 0.579·5-s − 6.53·6-s + 1.34·7-s − 7.07·8-s + 10/3·9-s + 1.63·10-s + 0.412·11-s + 11.5·12-s − 0.821·13-s − 3.79·14-s − 1.33·15-s + 35/4·16-s + 0.635·17-s − 9.42·18-s + 2.49·19-s − 2.89·20-s + 3.09·21-s − 1.16·22-s + 0.834·23-s − 16.3·24-s − 0.599·25-s + 2.32·26-s + 3.84·27-s + 6.71·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 23^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(9.225561502\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.225561502\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| 23 | $C_1$ | \( ( 1 - p^{3} T )^{4} \) |
| good | 5 | $C_2 \wr S_4$ | \( 1 + 162 T + 2924 p^{2} T^{2} + 176126 p^{3} T^{3} + 104115438 p^{3} T^{4} + 176126 p^{10} T^{5} + 2924 p^{16} T^{6} + 162 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 174 p T + 1560760 T^{2} - 187939214 p T^{3} + 1465644154542 T^{4} - 187939214 p^{8} T^{5} + 1560760 p^{14} T^{6} - 174 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 1820 T + 18076284 T^{2} + 135872642652 T^{3} - 298970183302154 T^{4} + 135872642652 p^{7} T^{5} + 18076284 p^{14} T^{6} - 1820 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 6508 T + 38067700 T^{2} + 156362285540 T^{3} + 2953677273591494 T^{4} + 156362285540 p^{7} T^{5} + 38067700 p^{14} T^{6} + 6508 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 12870 T + 827393180 T^{2} - 13926266271698 T^{3} + 376265033467995510 T^{4} - 13926266271698 p^{7} T^{5} + 827393180 p^{14} T^{6} - 12870 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 - 74474 T + 5460694192 T^{2} - 218107106860714 T^{3} + 432335047332304730 p T^{4} - 218107106860714 p^{7} T^{5} + 5460694192 p^{14} T^{6} - 74474 p^{21} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 94452 T + 59040800900 T^{2} + 4308011863456892 T^{3} + \)\(14\!\cdots\!74\)\( T^{4} + 4308011863456892 p^{7} T^{5} + 59040800900 p^{14} T^{6} + 94452 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 + 39420 T + 89436102348 T^{2} + 1950074313979404 T^{3} + \)\(34\!\cdots\!54\)\( T^{4} + 1950074313979404 p^{7} T^{5} + 89436102348 p^{14} T^{6} + 39420 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 75848 T + 280014961420 T^{2} - 12191087834241248 T^{3} + \)\(35\!\cdots\!34\)\( T^{4} - 12191087834241248 p^{7} T^{5} + 280014961420 p^{14} T^{6} - 75848 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 + 519032 T + 822460960460 T^{2} + 298427953268651592 T^{3} + \)\(24\!\cdots\!58\)\( T^{4} + 298427953268651592 p^{7} T^{5} + 822460960460 p^{14} T^{6} + 519032 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 716946 T + 763442776064 T^{2} - 11430701533568982 p T^{3} + \)\(26\!\cdots\!70\)\( T^{4} - 11430701533568982 p^{8} T^{5} + 763442776064 p^{14} T^{6} - 716946 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12232 T + 446860918604 T^{2} + 63322648731001896 T^{3} + \)\(38\!\cdots\!18\)\( T^{4} + 63322648731001896 p^{7} T^{5} + 446860918604 p^{14} T^{6} + 12232 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1053394 T + 4682460825892 T^{2} - 3579390387601538438 T^{3} + \)\(82\!\cdots\!18\)\( T^{4} - 3579390387601538438 p^{7} T^{5} + 4682460825892 p^{14} T^{6} - 1053394 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 4398344 T + 17112649414028 T^{2} - 37946485934999325768 T^{3} + \)\(74\!\cdots\!10\)\( T^{4} - 37946485934999325768 p^{7} T^{5} + 17112649414028 p^{14} T^{6} - 4398344 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 5328312 T + 20454975635708 T^{2} - 52675400124891059520 T^{3} + \)\(10\!\cdots\!66\)\( T^{4} - 52675400124891059520 p^{7} T^{5} + 20454975635708 p^{14} T^{6} - 5328312 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 11303966 T + 69619241267968 T^{2} - \)\(27\!\cdots\!02\)\( T^{3} + \)\(80\!\cdots\!82\)\( T^{4} - \)\(27\!\cdots\!02\)\( p^{7} T^{5} + 69619241267968 p^{14} T^{6} - 11303966 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 8395320 T + 55759031763980 T^{2} - \)\(23\!\cdots\!72\)\( T^{3} + \)\(83\!\cdots\!38\)\( T^{4} - \)\(23\!\cdots\!72\)\( p^{7} T^{5} + 55759031763980 p^{14} T^{6} - 8395320 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 - 7107008 T + 48359769293932 T^{2} - \)\(22\!\cdots\!28\)\( T^{3} + \)\(83\!\cdots\!22\)\( T^{4} - \)\(22\!\cdots\!28\)\( p^{7} T^{5} + 48359769293932 p^{14} T^{6} - 7107008 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 9785086 T + 104404988629832 T^{2} - \)\(59\!\cdots\!86\)\( T^{3} + \)\(32\!\cdots\!30\)\( T^{4} - \)\(59\!\cdots\!86\)\( p^{7} T^{5} + 104404988629832 p^{14} T^{6} - 9785086 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 2424316 T + 95769664514908 T^{2} - \)\(20\!\cdots\!72\)\( T^{3} + \)\(37\!\cdots\!78\)\( T^{4} - \)\(20\!\cdots\!72\)\( p^{7} T^{5} + 95769664514908 p^{14} T^{6} - 2424316 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 3019218 T + 122510228244116 T^{2} - \)\(54\!\cdots\!30\)\( T^{3} + \)\(68\!\cdots\!62\)\( T^{4} - \)\(54\!\cdots\!30\)\( p^{7} T^{5} + 122510228244116 p^{14} T^{6} - 3019218 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 32226876 T + 583197127116132 T^{2} - \)\(73\!\cdots\!08\)\( T^{3} + \)\(73\!\cdots\!74\)\( T^{4} - \)\(73\!\cdots\!08\)\( p^{7} T^{5} + 583197127116132 p^{14} T^{6} - 32226876 p^{21} T^{7} + p^{28} T^{8} \) |
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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
−8.345417988521852996040213653352, −7.972290365769898606928462559906, −7.901000958868625983169315633107, −7.61556201803272231801953429752, −7.60002799567682094426855024284, −7.19221916327566627603552687449, −6.86776064646190123784300382019, −6.59722316886562067375975930830, −6.53559094367019622429566924085, −5.48288393093949226331429334115, −5.37508666585563581216716427079, −5.12229580665165370744249419322, −4.83717270632665592016995588405, −3.88927683010727273945098771262, −3.62143155394359994126541624420, −3.55899039334449302988730037594, −3.39094573359874693445491391793, −2.44261089754871181480670149440, −2.34858866005830067801600976821, −2.19831766456289812018293232334, −1.96418293327540197224196345164, −1.23990081113545917004334436514, −0.910630406070035627948969286132, −0.795335927442366299803554041900, −0.60137327235980860546838446735,
0.60137327235980860546838446735, 0.795335927442366299803554041900, 0.910630406070035627948969286132, 1.23990081113545917004334436514, 1.96418293327540197224196345164, 2.19831766456289812018293232334, 2.34858866005830067801600976821, 2.44261089754871181480670149440, 3.39094573359874693445491391793, 3.55899039334449302988730037594, 3.62143155394359994126541624420, 3.88927683010727273945098771262, 4.83717270632665592016995588405, 5.12229580665165370744249419322, 5.37508666585563581216716427079, 5.48288393093949226331429334115, 6.53559094367019622429566924085, 6.59722316886562067375975930830, 6.86776064646190123784300382019, 7.19221916327566627603552687449, 7.60002799567682094426855024284, 7.61556201803272231801953429752, 7.901000958868625983169315633107, 7.972290365769898606928462559906, 8.345417988521852996040213653352
