| L(s) = 1 | + 4·9-s − 120·11-s + 152·19-s + 328·29-s − 1.51e3·31-s − 352·41-s + 1.70e3·49-s + 288·59-s − 2.07e3·61-s + 2.88e3·71-s + 2.11e3·79-s − 1.37e3·81-s + 3.30e3·89-s − 480·99-s + 3.12e3·101-s − 4.37e3·109-s + 6.16e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.21e4·169-s + ⋯ |
| L(s) = 1 | + 4/27·9-s − 3.28·11-s + 1.83·19-s + 2.10·29-s − 8.76·31-s − 1.34·41-s + 4.96·49-s + 0.635·59-s − 4.34·61-s + 4.82·71-s + 3.00·79-s − 1.88·81-s + 3.93·89-s − 0.487·99-s + 3.08·101-s − 3.84·109-s + 4.63·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 5.51·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{16} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.02899107481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02899107481\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( ( 1 - p T )^{8} \) |
| good | 3 | \( 1 - 4 T^{2} + 1388 T^{4} + 20420 T^{6} + 825814 T^{8} + 20420 p^{6} T^{10} + 1388 p^{12} T^{12} - 4 p^{18} T^{14} + p^{24} T^{16} \) |
| 7 | \( 1 - 1704 T^{2} + 1171836 T^{4} - 449061272 T^{6} + 142168007622 T^{8} - 449061272 p^{6} T^{10} + 1171836 p^{12} T^{12} - 1704 p^{18} T^{14} + p^{24} T^{16} \) |
| 11 | \( ( 1 + 60 T + 2316 T^{2} + 113340 T^{3} + 4880486 T^{4} + 113340 p^{3} T^{5} + 2316 p^{6} T^{6} + 60 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 13 | \( 1 - 932 p T^{2} + 70275916 T^{4} - 258212270572 T^{6} + 666683256664598 T^{8} - 258212270572 p^{6} T^{10} + 70275916 p^{12} T^{12} - 932 p^{19} T^{14} + p^{24} T^{16} \) |
| 17 | \( 1 - 22104 T^{2} + 244777436 T^{4} - 1846461785192 T^{6} + 36014987391878 p^{2} T^{8} - 1846461785192 p^{6} T^{10} + 244777436 p^{12} T^{12} - 22104 p^{18} T^{14} + p^{24} T^{16} \) |
| 23 | \( 1 - 32264 T^{2} + 739730300 T^{4} - 11600494415928 T^{6} + 301697119625702 p^{2} T^{8} - 11600494415928 p^{6} T^{10} + 739730300 p^{12} T^{12} - 32264 p^{18} T^{14} + p^{24} T^{16} \) |
| 29 | \( ( 1 - 164 T + 67428 T^{2} - 7628748 T^{3} + 2108741542 T^{4} - 7628748 p^{3} T^{5} + 67428 p^{6} T^{6} - 164 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 31 | \( ( 1 + 756 T + 284652 T^{2} + 70682020 T^{3} + 13533306150 T^{4} + 70682020 p^{3} T^{5} + 284652 p^{6} T^{6} + 756 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 37 | \( 1 - 4068 p T^{2} + 14307317708 T^{4} - 1042277057688780 T^{6} + 58185228077307981654 T^{8} - 1042277057688780 p^{6} T^{10} + 14307317708 p^{12} T^{12} - 4068 p^{19} T^{14} + p^{24} T^{16} \) |
| 41 | \( ( 1 + 176 T + 4860 T^{2} - 2023824 T^{3} + 2839160550 T^{4} - 2023824 p^{3} T^{5} + 4860 p^{6} T^{6} + 176 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 43 | \( 1 - 205384 T^{2} + 551185748 p T^{4} - 1456924152529976 T^{6} + 97664741956812577638 T^{8} - 1456924152529976 p^{6} T^{10} + 551185748 p^{13} T^{12} - 205384 p^{18} T^{14} + p^{24} T^{16} \) |
| 47 | \( 1 - 391432 T^{2} + 94374374012 T^{4} - 15190183527887544 T^{6} + \)\(18\!\cdots\!38\)\( T^{8} - 15190183527887544 p^{6} T^{10} + 94374374012 p^{12} T^{12} - 391432 p^{18} T^{14} + p^{24} T^{16} \) |
| 53 | \( 1 - 879108 T^{2} + 366543286508 T^{4} - 1793061386773900 p T^{6} + \)\(16\!\cdots\!26\)\( T^{8} - 1793061386773900 p^{7} T^{10} + 366543286508 p^{12} T^{12} - 879108 p^{18} T^{14} + p^{24} T^{16} \) |
| 59 | \( ( 1 - 144 T + 120012 T^{2} - 84322160 T^{3} + 24350475766 T^{4} - 84322160 p^{3} T^{5} + 120012 p^{6} T^{6} - 144 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 61 | \( ( 1 + 1036 T + 1266956 T^{2} + 748750628 T^{3} + 472708741350 T^{4} + 748750628 p^{3} T^{5} + 1266956 p^{6} T^{6} + 1036 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 67 | \( 1 - 729892 T^{2} + 350672969516 T^{4} - 113937688398727004 T^{6} + \)\(35\!\cdots\!18\)\( T^{8} - 113937688398727004 p^{6} T^{10} + 350672969516 p^{12} T^{12} - 729892 p^{18} T^{14} + p^{24} T^{16} \) |
| 71 | \( ( 1 - 1444 T + 1561404 T^{2} - 1265426996 T^{3} + 892485629222 T^{4} - 1265426996 p^{3} T^{5} + 1561404 p^{6} T^{6} - 1444 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 73 | \( 1 - 1034328 T^{2} + 11504470844 p T^{4} - 463742163140586216 T^{6} + \)\(20\!\cdots\!58\)\( T^{8} - 463742163140586216 p^{6} T^{10} + 11504470844 p^{13} T^{12} - 1034328 p^{18} T^{14} + p^{24} T^{16} \) |
| 79 | \( ( 1 - 1056 T + 1681052 T^{2} - 1307526656 T^{3} + 1197827670470 T^{4} - 1307526656 p^{3} T^{5} + 1681052 p^{6} T^{6} - 1056 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 83 | \( 1 - 325896 T^{2} + 2976058612 p T^{4} - 250700697356654072 T^{6} + \)\(11\!\cdots\!98\)\( T^{8} - 250700697356654072 p^{6} T^{10} + 2976058612 p^{13} T^{12} - 325896 p^{18} T^{14} + p^{24} T^{16} \) |
| 89 | \( ( 1 - 1652 T + 3468516 T^{2} - 3584681964 T^{3} + 3900476532118 T^{4} - 3584681964 p^{3} T^{5} + 3468516 p^{6} T^{6} - 1652 p^{9} T^{7} + p^{12} T^{8} )^{2} \) |
| 97 | \( 1 - 2532676 T^{2} + 3675970406668 T^{4} - 3589854946453003100 T^{6} + \)\(31\!\cdots\!54\)\( T^{8} - 3589854946453003100 p^{6} T^{10} + 3675970406668 p^{12} T^{12} - 2532676 p^{18} T^{14} + p^{24} T^{16} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{16} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.49454313745983486430478119873, −3.34546508596324567827279730530, −3.20585892971494000324348886787, −3.11423179004505705120187738241, −3.01291522559099422539688102202, −2.86455383405899596184547952392, −2.58075010371768169216284380926, −2.52547586581314135773181569608, −2.49297712178318973454209836824, −2.42514137435070644493040131195, −2.26004793240319607588932854654, −2.06460793215482633928338214834, −1.89787622685362991361418373479, −1.74734541943817494745847362578, −1.72209028689193745226658443399, −1.54164527180059794692973006721, −1.53919371219248210743627696514, −1.20836154742976919174889557013, −0.842103555924877099688203453524, −0.832345925331664600563991728671, −0.74999965804584864476180505694, −0.58991304244967209204896675574, −0.43071949026436465107722143205, −0.06347184548428527506320407734, −0.04167756952402515330121915201,
0.04167756952402515330121915201, 0.06347184548428527506320407734, 0.43071949026436465107722143205, 0.58991304244967209204896675574, 0.74999965804584864476180505694, 0.832345925331664600563991728671, 0.842103555924877099688203453524, 1.20836154742976919174889557013, 1.53919371219248210743627696514, 1.54164527180059794692973006721, 1.72209028689193745226658443399, 1.74734541943817494745847362578, 1.89787622685362991361418373479, 2.06460793215482633928338214834, 2.26004793240319607588932854654, 2.42514137435070644493040131195, 2.49297712178318973454209836824, 2.52547586581314135773181569608, 2.58075010371768169216284380926, 2.86455383405899596184547952392, 3.01291522559099422539688102202, 3.11423179004505705120187738241, 3.20585892971494000324348886787, 3.34546508596324567827279730530, 3.49454313745983486430478119873
Plot not available for L-functions of degree greater than 10.
