| L(s) = 1 | − 4·2-s + 6·4-s + 2·7-s + 2·9-s + 6·11-s + 6·13-s − 8·14-s − 15·16-s − 8·18-s + 6·19-s − 24·22-s + 6·23-s − 24·26-s + 12·28-s + 8·29-s + 24·32-s + 12·36-s − 10·37-s − 24·38-s − 48·43-s + 36·44-s − 24·46-s − 16·47-s + 9·49-s + 36·52-s − 32·58-s − 24·59-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 3·4-s + 0.755·7-s + 2/3·9-s + 1.80·11-s + 1.66·13-s − 2.13·14-s − 3.75·16-s − 1.88·18-s + 1.37·19-s − 5.11·22-s + 1.25·23-s − 4.70·26-s + 2.26·28-s + 1.48·29-s + 4.24·32-s + 2·36-s − 1.64·37-s − 3.89·38-s − 7.31·43-s + 5.42·44-s − 3.53·46-s − 2.33·47-s + 9/7·49-s + 4.99·52-s − 4.20·58-s − 3.12·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 5^{16} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.01435392824\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01435392824\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T + T^{2} )^{4} \) |
| 3 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 6 T + 45 T^{2} - 186 T^{3} + 848 T^{4} - 186 p T^{5} + 45 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 - 2 T - 5 T^{2} + 26 T^{3} - 43 T^{4} - 32 T^{5} + 338 T^{6} - 556 T^{7} - 614 T^{8} - 556 p T^{9} + 338 p^{2} T^{10} - 32 p^{3} T^{11} - 43 p^{4} T^{12} + 26 p^{5} T^{13} - 5 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 6 T + 26 T^{2} - 84 T^{3} + 93 T^{4} - 216 T^{5} + 214 T^{6} + 366 T^{7} + 9716 T^{8} + 366 p T^{9} + 214 p^{2} T^{10} - 216 p^{3} T^{11} + 93 p^{4} T^{12} - 84 p^{5} T^{13} + 26 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 18 T^{2} + 35 T^{4} + 18 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 6 T + 31 T^{2} - 6 p T^{3} + 217 T^{4} - 2234 T^{6} + 6492 T^{7} - 40022 T^{8} + 6492 p T^{9} - 2234 p^{2} T^{10} + 217 p^{4} T^{12} - 6 p^{6} T^{13} + 31 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 - 6 T + 59 T^{2} - 282 T^{3} + 1905 T^{4} - 10224 T^{5} + 45502 T^{6} - 270276 T^{7} + 987242 T^{8} - 270276 p T^{9} + 45502 p^{2} T^{10} - 10224 p^{3} T^{11} + 1905 p^{4} T^{12} - 282 p^{5} T^{13} + 59 p^{6} T^{14} - 6 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 8 T - 13 T^{2} + 480 T^{3} - 1615 T^{4} - 7528 T^{5} + 59362 T^{6} + 2520 T^{7} - 1294994 T^{8} + 2520 p T^{9} + 59362 p^{2} T^{10} - 7528 p^{3} T^{11} - 1615 p^{4} T^{12} + 480 p^{5} T^{13} - 13 p^{6} T^{14} - 8 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 74 T^{2} + 3321 T^{4} - 90346 T^{6} + 2694452 T^{8} - 90346 p^{2} T^{10} + 3321 p^{4} T^{12} - 74 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( 1 + 10 T - 14 T^{2} - 220 T^{3} + 1301 T^{4} + 4780 T^{5} - 35098 T^{6} - 340690 T^{7} - 2682860 T^{8} - 340690 p T^{9} - 35098 p^{2} T^{10} + 4780 p^{3} T^{11} + 1301 p^{4} T^{12} - 220 p^{5} T^{13} - 14 p^{6} T^{14} + 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( 1 + 80 T^{2} + 2370 T^{4} - 7488 T^{5} + 65728 T^{6} - 773760 T^{7} + 3160547 T^{8} - 773760 p T^{9} + 65728 p^{2} T^{10} - 7488 p^{3} T^{11} + 2370 p^{4} T^{12} + 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 43 | \( 1 + 48 T + 1183 T^{2} + 19920 T^{3} + 256773 T^{4} + 2696256 T^{5} + 558818 p T^{6} + 187083744 T^{7} + 1296211046 T^{8} + 187083744 p T^{9} + 558818 p^{3} T^{10} + 2696256 p^{3} T^{11} + 256773 p^{4} T^{12} + 19920 p^{5} T^{13} + 1183 p^{6} T^{14} + 48 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( ( 1 + 8 T + 106 T^{2} + 352 T^{3} + 4443 T^{4} + 352 p T^{5} + 106 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 53 | \( 1 - 210 T^{2} + 16393 T^{4} - 540522 T^{6} + 11949540 T^{8} - 540522 p^{2} T^{10} + 16393 p^{4} T^{12} - 210 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 + 24 T + 475 T^{2} + 6792 T^{3} + 87253 T^{4} + 932736 T^{5} + 9243862 T^{6} + 80308224 T^{7} + 654943486 T^{8} + 80308224 p T^{9} + 9243862 p^{2} T^{10} + 932736 p^{3} T^{11} + 87253 p^{4} T^{12} + 6792 p^{5} T^{13} + 475 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( ( 1 + 8 T - 62 T^{2} + 32 T^{3} + 8251 T^{4} + 32 p T^{5} - 62 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 12 T - 72 T^{2} - 1944 T^{3} - 878 T^{4} + 154212 T^{5} + 870912 T^{6} - 4319124 T^{7} - 74785245 T^{8} - 4319124 p T^{9} + 870912 p^{2} T^{10} + 154212 p^{3} T^{11} - 878 p^{4} T^{12} - 1944 p^{5} T^{13} - 72 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( 1 + 12 T + 128 T^{2} + 960 T^{3} + 5202 T^{4} - 11124 T^{5} - 245984 T^{6} - 4504236 T^{7} - 41845549 T^{8} - 4504236 p T^{9} - 245984 p^{2} T^{10} - 11124 p^{3} T^{11} + 5202 p^{4} T^{12} + 960 p^{5} T^{13} + 128 p^{6} T^{14} + 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 73 | \( ( 1 - 12 T + 168 T^{2} - 900 T^{3} + 9326 T^{4} - 900 p T^{5} + 168 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 10 T + 169 T^{2} - 1510 T^{3} + 17728 T^{4} - 1510 p T^{5} + 169 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 83 | \( ( 1 + 16 T + 232 T^{2} + 2048 T^{3} + 19710 T^{4} + 2048 p T^{5} + 232 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 + 42 T + 1115 T^{2} + 22134 T^{3} + 362637 T^{4} + 5064156 T^{5} + 62362882 T^{6} + 684879408 T^{7} + 6796999778 T^{8} + 684879408 p T^{9} + 62362882 p^{2} T^{10} + 5064156 p^{3} T^{11} + 362637 p^{4} T^{12} + 22134 p^{5} T^{13} + 1115 p^{6} T^{14} + 42 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( 1 - 36 T + 528 T^{2} - 5448 T^{3} + 65458 T^{4} - 618348 T^{5} + 3080736 T^{6} - 24087660 T^{7} + 342649875 T^{8} - 24087660 p T^{9} + 3080736 p^{2} T^{10} - 618348 p^{3} T^{11} + 65458 p^{4} T^{12} - 5448 p^{5} T^{13} + 528 p^{6} T^{14} - 36 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−3.81707990571778229703572103345, −3.75383324249149748895611571784, −3.60607432446953567035895247042, −3.42056017954256725643326753786, −3.35684352014078973626308151218, −3.28936711293253176984669199846, −2.96373516516642412627754848148, −2.95747482854840611284755990354, −2.93999965875440290276380987972, −2.73856732174122696160938306624, −2.68791264755028670974620170878, −2.38137917863710547163422267530, −2.02540797452205597565317472284, −1.78130133091470034467579794073, −1.76915723045215910315676390544, −1.63049433138051925642494887692, −1.56744906172517292990162975616, −1.52480289663271850324816430095, −1.45899157530161177186398045257, −1.32110351723315694892945962564, −0.943511795869769830843641518306, −0.889686833311156234120951314988, −0.828398917652786137410891022377, −0.17791436845095923230113868851, −0.04243813565378289368779543545,
0.04243813565378289368779543545, 0.17791436845095923230113868851, 0.828398917652786137410891022377, 0.889686833311156234120951314988, 0.943511795869769830843641518306, 1.32110351723315694892945962564, 1.45899157530161177186398045257, 1.52480289663271850324816430095, 1.56744906172517292990162975616, 1.63049433138051925642494887692, 1.76915723045215910315676390544, 1.78130133091470034467579794073, 2.02540797452205597565317472284, 2.38137917863710547163422267530, 2.68791264755028670974620170878, 2.73856732174122696160938306624, 2.93999965875440290276380987972, 2.95747482854840611284755990354, 2.96373516516642412627754848148, 3.28936711293253176984669199846, 3.35684352014078973626308151218, 3.42056017954256725643326753786, 3.60607432446953567035895247042, 3.75383324249149748895611571784, 3.81707990571778229703572103345
Plot not available for L-functions of degree greater than 10.
