| L(s) = 1 | + 16·2-s + 64·4-s + 455·5-s + 1.55e3·7-s − 1.02e3·8-s + 7.28e3·10-s − 1.17e3·11-s − 4.09e4·13-s + 2.48e4·14-s − 1.63e4·16-s − 3.96e3·17-s + 3.05e4·19-s + 2.91e4·20-s − 1.88e4·22-s + 9.96e4·23-s + 1.73e5·25-s − 6.54e5·26-s + 9.94e4·28-s − 3.28e5·29-s − 1.08e5·31-s − 6.55e4·32-s − 6.33e4·34-s + 7.07e5·35-s + 2.59e5·37-s + 4.89e5·38-s − 4.65e5·40-s − 2.46e6·41-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1/2·4-s + 1.62·5-s + 1.71·7-s − 0.707·8-s + 2.30·10-s − 0.266·11-s − 5.16·13-s + 2.42·14-s − 16-s − 0.195·17-s + 1.02·19-s + 0.813·20-s − 0.376·22-s + 1.70·23-s + 2.22·25-s − 7.30·26-s + 0.856·28-s − 2.50·29-s − 0.651·31-s − 0.353·32-s − 0.276·34-s + 2.78·35-s + 0.843·37-s + 1.44·38-s − 1.15·40-s − 5.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(8-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+7/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.04863983935\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04863983935\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T + p^{6} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2^2$ | \( 1 - 222 p T + 3421 p^{3} T^{2} - 222 p^{8} T^{3} + p^{14} T^{4} \) |
| good | 5 | $D_4\times C_2$ | \( 1 - 91 p T + 267 p^{3} T^{2} - 63336 p^{3} T^{3} + 12603166 p^{4} T^{4} - 63336 p^{10} T^{5} + 267 p^{17} T^{6} - 91 p^{22} T^{7} + p^{28} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 + 1175 T - 26558709 T^{2} - 12966134400 T^{3} + 373952075318440 T^{4} - 12966134400 p^{7} T^{5} - 26558709 p^{14} T^{6} + 1175 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 20461 T + 208687508 T^{2} + 20461 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 + 3962 T + 137816562 T^{2} - 3735359590368 T^{3} - 159906015918754817 T^{4} - 3735359590368 p^{7} T^{5} + 137816562 p^{14} T^{6} + 3962 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 - 30569 T + 45392203 p T^{2} + 52448198485606 T^{3} - 1629111718542729284 T^{4} + 52448198485606 p^{7} T^{5} + 45392203 p^{15} T^{6} - 30569 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 4334 p T + 80515242 p T^{2} - 240254284512 p^{2} T^{3} + 37687497994880263 p^{2} T^{4} - 240254284512 p^{9} T^{5} + 80515242 p^{15} T^{6} - 4334 p^{22} T^{7} + p^{28} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 164357 T + 18365528026 T^{2} + 164357 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 108038 T - 4457788831 T^{2} - 135553640419806 p T^{3} - 851092700442076180 p^{2} T^{4} - 135553640419806 p^{8} T^{5} - 4457788831 p^{14} T^{6} + 108038 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 259979 T - 138996797147 T^{2} - 4347400899130238 T^{3} + \)\(27\!\cdots\!02\)\( T^{4} - 4347400899130238 p^{7} T^{5} - 138996797147 p^{14} T^{6} - 259979 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 1231944 T + 721416183346 T^{2} + 1231944 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 1205387 T + 833903495058 T^{2} + 1205387 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 + 1426068 T + 878565721874 T^{2} + 202299122719516032 T^{3} + \)\(47\!\cdots\!19\)\( T^{4} + 202299122719516032 p^{7} T^{5} + 878565721874 p^{14} T^{6} + 1426068 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 + 1095975 T - 329259768721 T^{2} - 897604961086729800 T^{3} - \)\(44\!\cdots\!78\)\( T^{4} - 897604961086729800 p^{7} T^{5} - 329259768721 p^{14} T^{6} + 1095975 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 + 1503103 T - 942513949773 T^{2} - 2668714871508067368 T^{3} - \)\(15\!\cdots\!08\)\( T^{4} - 2668714871508067368 p^{7} T^{5} - 942513949773 p^{14} T^{6} + 1503103 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 2512006 T - 503121772798 T^{2} - 1325862501778972752 T^{3} + \)\(16\!\cdots\!59\)\( T^{4} - 1325862501778972752 p^{7} T^{5} - 503121772798 p^{14} T^{6} - 2512006 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 4713159 T + 5017229011421 T^{2} - 23920297786159999026 T^{3} + \)\(12\!\cdots\!08\)\( T^{4} - 23920297786159999026 p^{7} T^{5} + 5017229011421 p^{14} T^{6} - 4713159 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 + 5779648 T + 17777757291310 T^{2} + 5779648 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 5174421 T - 1942363556779 T^{2} - 34266046626460534746 T^{3} + \)\(40\!\cdots\!14\)\( T^{4} - 34266046626460534746 p^{7} T^{5} - 1942363556779 p^{14} T^{6} - 5174421 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $D_4\times C_2$ | \( 1 - 8656036 T + 20722433621807 T^{2} - \)\(13\!\cdots\!56\)\( T^{3} + \)\(11\!\cdots\!76\)\( T^{4} - \)\(13\!\cdots\!56\)\( p^{7} T^{5} + 20722433621807 p^{14} T^{6} - 8656036 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 3738203 T + 56498371315198 T^{2} - 3738203 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
| 89 | $D_4\times C_2$ | \( 1 + 5013414 T - 60764320556494 T^{2} - 12854540384859039552 T^{3} + \)\(43\!\cdots\!75\)\( T^{4} - 12854540384859039552 p^{7} T^{5} - 60764320556494 p^{14} T^{6} + 5013414 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 6732033 T + 142242614798092 T^{2} - 6732033 p^{7} T^{3} + p^{14} T^{4} )^{2} \) |
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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
−8.464372268705843246707617523598, −8.233914325996822165952541665771, −7.70838477438333278662644515413, −7.32612250250890324129825703887, −7.31624795509883951273284075496, −6.83133711955175445225885894360, −6.67358988512996792150519318938, −6.57701748429850289864431794198, −5.61054724034898557535408149250, −5.52778684830168678103080396532, −5.16753839282786716201671425679, −4.94797757608040554092604204310, −4.85553763071110898221704529296, −4.78199928829756533284522102031, −4.75317159437619247964608408273, −3.55389533790806516528465923646, −3.28087930122593644733015429205, −3.25268674144960962070359301959, −2.65394783592693809665113447287, −2.23768683336383998038285995919, −1.90886554450363646977141921338, −1.67874628641140422351566045115, −1.57889035824928151986570884094, −0.51230132305427476144738950526, −0.02464986410689536463263449482,
0.02464986410689536463263449482, 0.51230132305427476144738950526, 1.57889035824928151986570884094, 1.67874628641140422351566045115, 1.90886554450363646977141921338, 2.23768683336383998038285995919, 2.65394783592693809665113447287, 3.25268674144960962070359301959, 3.28087930122593644733015429205, 3.55389533790806516528465923646, 4.75317159437619247964608408273, 4.78199928829756533284522102031, 4.85553763071110898221704529296, 4.94797757608040554092604204310, 5.16753839282786716201671425679, 5.52778684830168678103080396532, 5.61054724034898557535408149250, 6.57701748429850289864431794198, 6.67358988512996792150519318938, 6.83133711955175445225885894360, 7.31624795509883951273284075496, 7.32612250250890324129825703887, 7.70838477438333278662644515413, 8.233914325996822165952541665771, 8.464372268705843246707617523598
