| L(s) = 1 | + 104·5-s + 492·7-s + 2.10e3·11-s + 2.35e3·13-s − 4.13e3·17-s − 5.51e3·19-s + 1.78e4·23-s − 1.17e5·25-s + 1.50e5·29-s + 7.82e4·31-s + 5.11e4·35-s − 4.23e4·37-s + 2.80e5·41-s + 5.12e4·43-s − 8.29e5·47-s − 1.51e6·49-s + 1.76e6·53-s + 2.18e5·55-s − 1.88e6·59-s + 8.69e5·61-s + 2.45e5·65-s + 2.49e5·67-s − 9.34e6·71-s − 4.61e5·73-s + 1.03e6·77-s + 9.36e5·79-s − 1.58e7·83-s + ⋯ |
| L(s) = 1 | + 0.372·5-s + 0.542·7-s + 0.476·11-s + 0.297·13-s − 0.204·17-s − 0.184·19-s + 0.305·23-s − 1.50·25-s + 1.14·29-s + 0.471·31-s + 0.201·35-s − 0.137·37-s + 0.636·41-s + 0.0982·43-s − 1.16·47-s − 1.84·49-s + 1.62·53-s + 0.177·55-s − 1.19·59-s + 0.490·61-s + 0.110·65-s + 0.101·67-s − 3.09·71-s − 0.138·73-s + 0.258·77-s + 0.213·79-s − 3.03·83-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
| \(L(4)\) |
\(\approx\) |
\(10.18476913\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.18476913\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2 \wr S_4$ | \( 1 - 104 T + 25684 p T^{2} - 13432 p^{5} T^{3} + 61198862 p^{3} T^{4} - 13432 p^{12} T^{5} + 25684 p^{15} T^{6} - 104 p^{21} T^{7} + p^{28} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 492 T + 1761490 T^{2} - 884061288 T^{3} + 1569188693355 T^{4} - 884061288 p^{7} T^{5} + 1761490 p^{14} T^{6} - 492 p^{21} T^{7} + p^{28} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 - 2104 T + 24416636 T^{2} + 16337041928 T^{3} + 409349654705878 T^{4} + 16337041928 p^{7} T^{5} + 24416636 p^{14} T^{6} - 2104 p^{21} T^{7} + p^{28} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - 2356 T + 74449642 T^{2} - 506957513632 T^{3} + 6209714322557923 T^{4} - 506957513632 p^{7} T^{5} + 74449642 p^{14} T^{6} - 2356 p^{21} T^{7} + p^{28} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 4136 T + 27639412 p T^{2} + 3070746324728 T^{3} + 159177270860618278 T^{4} + 3070746324728 p^{7} T^{5} + 27639412 p^{15} T^{6} + 4136 p^{21} T^{7} + p^{28} T^{8} \) |
| 19 | $C_2 \wr S_4$ | \( 1 + 5516 T + 964110754 T^{2} + 16814083801016 T^{3} + 1016867687399988811 T^{4} + 16814083801016 p^{7} T^{5} + 964110754 p^{14} T^{6} + 5516 p^{21} T^{7} + p^{28} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 - 776 p T + 10459747340 T^{2} - 149693197603096 T^{3} + 49659200356592362438 T^{4} - 149693197603096 p^{7} T^{5} + 10459747340 p^{14} T^{6} - 776 p^{22} T^{7} + p^{28} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 - 150720 T + 19284779348 T^{2} + 3190673772727488 T^{3} - \)\(46\!\cdots\!38\)\( T^{4} + 3190673772727488 p^{7} T^{5} + 19284779348 p^{14} T^{6} - 150720 p^{21} T^{7} + p^{28} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 78256 T + 38982228700 T^{2} - 3220005537582064 T^{3} + \)\(92\!\cdots\!54\)\( T^{4} - 3220005537582064 p^{7} T^{5} + 38982228700 p^{14} T^{6} - 78256 p^{21} T^{7} + p^{28} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 42324 T + 220774121194 T^{2} + 26921824413003072 T^{3} + \)\(25\!\cdots\!83\)\( T^{4} + 26921824413003072 p^{7} T^{5} + 220774121194 p^{14} T^{6} + 42324 p^{21} T^{7} + p^{28} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 280704 T + 472877427044 T^{2} - 112603271161617792 T^{3} + \)\(12\!\cdots\!06\)\( T^{4} - 112603271161617792 p^{7} T^{5} + 472877427044 p^{14} T^{6} - 280704 p^{21} T^{7} + p^{28} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 51200 T + 984446159884 T^{2} - 48153078330116096 T^{3} + \)\(38\!\cdots\!30\)\( T^{4} - 48153078330116096 p^{7} T^{5} + 984446159884 p^{14} T^{6} - 51200 p^{21} T^{7} + p^{28} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 829608 T + 1409037458732 T^{2} + 1196006982169373064 T^{3} + \)\(96\!\cdots\!78\)\( T^{4} + 1196006982169373064 p^{7} T^{5} + 1409037458732 p^{14} T^{6} + 829608 p^{21} T^{7} + p^{28} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 - 1763120 T + 3909373559732 T^{2} - 4237195044933381968 T^{3} + \)\(60\!\cdots\!02\)\( T^{4} - 4237195044933381968 p^{7} T^{5} + 3909373559732 p^{14} T^{6} - 1763120 p^{21} T^{7} + p^{28} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 + 1881992 T + 4417947254204 T^{2} + 6866322179826846536 T^{3} + \)\(16\!\cdots\!62\)\( T^{4} + 6866322179826846536 p^{7} T^{5} + 4417947254204 p^{14} T^{6} + 1881992 p^{21} T^{7} + p^{28} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 869116 T + 1883044600378 T^{2} + 709665817077998912 T^{3} + \)\(11\!\cdots\!19\)\( T^{4} + 709665817077998912 p^{7} T^{5} + 1883044600378 p^{14} T^{6} - 869116 p^{21} T^{7} + p^{28} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 249700 T + 22505599581394 T^{2} - 4689052392598157320 T^{3} + \)\(19\!\cdots\!27\)\( T^{4} - 4689052392598157320 p^{7} T^{5} + 22505599581394 p^{14} T^{6} - 249700 p^{21} T^{7} + p^{28} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 9344848 T + 61160605397660 T^{2} + \)\(26\!\cdots\!52\)\( T^{3} + \)\(92\!\cdots\!78\)\( T^{4} + \)\(26\!\cdots\!52\)\( p^{7} T^{5} + 61160605397660 p^{14} T^{6} + 9344848 p^{21} T^{7} + p^{28} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 461660 T + 23316950476234 T^{2} - 32405975938074761584 T^{3} + \)\(24\!\cdots\!95\)\( T^{4} - 32405975938074761584 p^{7} T^{5} + 23316950476234 p^{14} T^{6} + 461660 p^{21} T^{7} + p^{28} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 936116 T + 14125575453586 T^{2} + 80420170599094260040 T^{3} - \)\(26\!\cdots\!29\)\( T^{4} + 80420170599094260040 p^{7} T^{5} + 14125575453586 p^{14} T^{6} - 936116 p^{21} T^{7} + p^{28} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 15826384 T + 178084288814252 T^{2} + \)\(14\!\cdots\!28\)\( T^{3} + \)\(82\!\cdots\!78\)\( T^{4} + \)\(14\!\cdots\!28\)\( p^{7} T^{5} + 178084288814252 p^{14} T^{6} + 15826384 p^{21} T^{7} + p^{28} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 22823736 T + 324355782855956 T^{2} - \)\(30\!\cdots\!88\)\( T^{3} + \)\(23\!\cdots\!90\)\( T^{4} - \)\(30\!\cdots\!88\)\( p^{7} T^{5} + 324355782855956 p^{14} T^{6} - 22823736 p^{21} T^{7} + p^{28} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 - 9596660 T + 291341934016954 T^{2} - \)\(19\!\cdots\!52\)\( T^{3} + \)\(33\!\cdots\!31\)\( T^{4} - \)\(19\!\cdots\!52\)\( p^{7} T^{5} + 291341934016954 p^{14} T^{6} - 9596660 p^{21} T^{7} + p^{28} T^{8} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.84964265549554977735057342392, −6.39853042721568366978428121665, −6.28128114356101691750029511050, −6.20314184171773519077929062261, −6.14987124574417047937125813515, −5.39991241607789343635789583474, −5.36218189603456539898778022066, −5.33248854407954496742955281898, −4.80994548257802845196502440692, −4.44689593665931874164603619806, −4.30432441445703748889081368678, −4.15264242585801901591029393648, −4.01089532889130371579139370678, −3.17771796550859574688562356326, −3.16085054699602505331867057873, −3.06161608568051968398792167476, −2.84663434883004745634203529431, −1.99275734558051534311559880620, −1.93327107518229313634059527561, −1.78807158637428438233715103044, −1.72612901735057319835158277499, −1.02721531521428437068559787252, −0.73280189872720113192693733028, −0.57106689184779685538952289704, −0.33414357540699778441641663664,
0.33414357540699778441641663664, 0.57106689184779685538952289704, 0.73280189872720113192693733028, 1.02721531521428437068559787252, 1.72612901735057319835158277499, 1.78807158637428438233715103044, 1.93327107518229313634059527561, 1.99275734558051534311559880620, 2.84663434883004745634203529431, 3.06161608568051968398792167476, 3.16085054699602505331867057873, 3.17771796550859574688562356326, 4.01089532889130371579139370678, 4.15264242585801901591029393648, 4.30432441445703748889081368678, 4.44689593665931874164603619806, 4.80994548257802845196502440692, 5.33248854407954496742955281898, 5.36218189603456539898778022066, 5.39991241607789343635789583474, 6.14987124574417047937125813515, 6.20314184171773519077929062261, 6.28128114356101691750029511050, 6.39853042721568366978428121665, 6.84964265549554977735057342392
