| L(s) = 1 | + 2·4-s − 6·11-s − 12·13-s + 16-s − 16·17-s − 6·19-s − 4·23-s − 4·25-s + 8·29-s + 30·37-s + 14·43-s − 12·44-s − 7·49-s − 24·52-s − 16·53-s − 24·59-s − 16·61-s − 2·64-s + 24·67-s − 32·68-s + 12·71-s − 12·76-s − 20·79-s − 42·89-s − 8·92-s − 24·97-s − 8·100-s + ⋯ |
| L(s) = 1 | + 4-s − 1.80·11-s − 3.32·13-s + 1/4·16-s − 3.88·17-s − 1.37·19-s − 0.834·23-s − 4/5·25-s + 1.48·29-s + 4.93·37-s + 2.13·43-s − 1.80·44-s − 49-s − 3.32·52-s − 2.19·53-s − 3.12·59-s − 2.04·61-s − 1/4·64-s + 2.93·67-s − 3.88·68-s + 1.42·71-s − 1.37·76-s − 2.25·79-s − 4.45·89-s − 0.834·92-s − 2.43·97-s − 4/5·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 5^{8} \cdot 13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.07621599877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.07621599877\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 13 | \( 1 + 12 T + 45 T^{2} - 24 T^{3} - 556 T^{4} - 24 p T^{5} + 45 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 + p T^{2} - 3 T^{4} - 130 T^{6} - 1248 T^{7} - 754 T^{8} - 1248 p T^{9} - 130 p^{2} T^{10} - 3 p^{4} T^{12} + p^{7} T^{14} + p^{8} T^{16} \) |
| 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 | \( ( 1 + 4 T - T^{2} + 4 p T^{3} + p^{2} T^{4} )^{4} \) |
| 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 | \( 1 + 4 T - 33 T^{2} - 324 T^{3} + 113 T^{4} + 8736 T^{5} + 34370 T^{6} - 109208 T^{7} - 1157922 T^{8} - 109208 p T^{9} + 34370 p^{2} T^{10} + 8736 p^{3} T^{11} + 113 p^{4} T^{12} - 324 p^{5} T^{13} - 33 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 - 30 T + 514 T^{2} - 6420 T^{3} + 64101 T^{4} - 541620 T^{5} + 4034198 T^{6} - 27314130 T^{7} + 171690740 T^{8} - 27314130 p T^{9} + 4034198 p^{2} T^{10} - 541620 p^{3} T^{11} + 64101 p^{4} T^{12} - 6420 p^{5} T^{13} + 514 p^{6} T^{14} - 30 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 - 14 T + 67 T^{2} + 134 T^{3} - 3763 T^{4} + 24232 T^{5} - 46294 T^{6} - 310492 T^{7} + 3087010 T^{8} - 310492 p T^{9} - 46294 p^{2} T^{10} + 24232 p^{3} T^{11} - 3763 p^{4} T^{12} + 134 p^{5} T^{13} + 67 p^{6} T^{14} - 14 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 148 T^{2} + 14490 T^{4} - 1021616 T^{6} + 54203939 T^{8} - 1021616 p^{2} T^{10} + 14490 p^{4} T^{12} - 148 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 8 T + 137 T^{2} + 1268 T^{3} + 8956 T^{4} + 1268 p T^{5} + 137 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 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 | \( 1 - 24 T + 432 T^{2} - 5760 T^{3} + 69970 T^{4} - 776424 T^{5} + 7801920 T^{6} - 72545640 T^{7} + 609479859 T^{8} - 72545640 p T^{9} + 7801920 p^{2} T^{10} - 776424 p^{3} T^{11} + 69970 p^{4} T^{12} - 5760 p^{5} T^{13} + 432 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 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 | \( 1 - 192 T^{2} + 25276 T^{4} - 2537280 T^{6} + 214285254 T^{8} - 2537280 p^{2} T^{10} + 25276 p^{4} T^{12} - 192 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( 1 - 208 T^{2} + 27708 T^{4} - 2708144 T^{6} + 235978406 T^{8} - 2708144 p^{2} T^{10} + 27708 p^{4} T^{12} - 208 p^{6} T^{14} + p^{8} T^{16} \) |
| 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 | \( 1 + 24 T + 408 T^{2} + 5184 T^{3} + 49906 T^{4} + 311304 T^{5} + 4896 T^{6} - 24434760 T^{7} - 344198589 T^{8} - 24434760 p T^{9} + 4896 p^{2} T^{10} + 311304 p^{3} T^{11} + 49906 p^{4} T^{12} + 5184 p^{5} T^{13} + 408 p^{6} T^{14} + 24 p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.23630950516548655963989687817, −4.19138868856853126226484810827, −4.10458087452454454105547224869, −3.94542360508794756095070422306, −3.73396721354986972488526500414, −3.64067014981547014206668391200, −3.02839691061777457433186930859, −2.92920981717755071749168568741, −2.88885764529529089628864547611, −2.87749098067409454654123833248, −2.86640319679756095697962114484, −2.78853039780930073150346358858, −2.61545850832663051529731483057, −2.42429505767596539199650804496, −2.20699698967182890680071520121, −2.14888817048056919264488047327, −2.00594911879120091550599121157, −1.95529948972394899007408723875, −1.71869645187532397674519956775, −1.46706815936026366543054908359, −1.15644177063562560081303099987, −1.12969542336146042673606271451, −0.45049431612109964127777416393, −0.33267565347274654296874131967, −0.06381862174671430237207036400,
0.06381862174671430237207036400, 0.33267565347274654296874131967, 0.45049431612109964127777416393, 1.12969542336146042673606271451, 1.15644177063562560081303099987, 1.46706815936026366543054908359, 1.71869645187532397674519956775, 1.95529948972394899007408723875, 2.00594911879120091550599121157, 2.14888817048056919264488047327, 2.20699698967182890680071520121, 2.42429505767596539199650804496, 2.61545850832663051529731483057, 2.78853039780930073150346358858, 2.86640319679756095697962114484, 2.87749098067409454654123833248, 2.88885764529529089628864547611, 2.92920981717755071749168568741, 3.02839691061777457433186930859, 3.64067014981547014206668391200, 3.73396721354986972488526500414, 3.94542360508794756095070422306, 4.10458087452454454105547224869, 4.19138868856853126226484810827, 4.23630950516548655963989687817
Plot not available for L-functions of degree greater than 10.
