| L(s) = 1 | + 4·3-s + 4·4-s − 4·5-s + 2·7-s − 2·8-s + 6·9-s + 2·11-s + 16·12-s + 12·13-s − 16·15-s + 9·16-s + 2·17-s − 4·19-s − 16·20-s + 8·21-s − 5·23-s − 8·24-s + 20·25-s + 8·28-s − 8·29-s − 17·31-s − 10·32-s + 8·33-s − 8·35-s + 24·36-s + 3·37-s + 48·39-s + ⋯ |
| L(s) = 1 | + 2.30·3-s + 2·4-s − 1.78·5-s + 0.755·7-s − 0.707·8-s + 2·9-s + 0.603·11-s + 4.61·12-s + 3.32·13-s − 4.13·15-s + 9/4·16-s + 0.485·17-s − 0.917·19-s − 3.57·20-s + 1.74·21-s − 1.04·23-s − 1.63·24-s + 4·25-s + 1.51·28-s − 1.48·29-s − 3.05·31-s − 1.76·32-s + 1.39·33-s − 1.35·35-s + 4·36-s + 0.493·37-s + 7.68·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{8} \cdot 7^{8} \cdot 19^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(12.84237155\) |
| \(L(\frac12)\) |
\(\approx\) |
\(12.84237155\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( ( 1 - T + T^{2} )^{4} \) |
| 7 | \( 1 - 2 T - 5 T^{2} + 8 T^{3} + 5 T^{4} + 8 p T^{5} - 5 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | \( ( 1 + T + T^{2} )^{4} \) |
| good | 2 | \( 1 - p^{2} T^{2} + p T^{3} + 7 T^{4} - 3 p T^{5} - 3 T^{6} + 7 T^{7} - T^{8} + 7 p T^{9} - 3 p^{2} T^{10} - 3 p^{4} T^{11} + 7 p^{4} T^{12} + p^{6} T^{13} - p^{8} T^{14} + p^{8} T^{16} \) |
| 5 | \( 1 + 4 T - 4 T^{2} - 26 T^{3} + 7 p T^{4} + 101 T^{5} - 311 T^{6} + 9 T^{7} + 2739 T^{8} + 9 p T^{9} - 311 p^{2} T^{10} + 101 p^{3} T^{11} + 7 p^{5} T^{12} - 26 p^{5} T^{13} - 4 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 11 | \( 1 - 2 T - 18 T^{2} + 142 T^{3} + 5 p T^{4} - 2097 T^{5} + 6581 T^{6} + 14427 T^{7} - 98647 T^{8} + 14427 p T^{9} + 6581 p^{2} T^{10} - 2097 p^{3} T^{11} + 5 p^{5} T^{12} + 142 p^{5} T^{13} - 18 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( ( 1 - 6 T + 42 T^{2} - 127 T^{3} + 613 T^{4} - 127 p T^{5} + 42 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( 1 - 2 T - 48 T^{2} + 64 T^{3} + 1299 T^{4} - 848 T^{5} - 30068 T^{6} + 4722 T^{7} + 584544 T^{8} + 4722 p T^{9} - 30068 p^{2} T^{10} - 848 p^{3} T^{11} + 1299 p^{4} T^{12} + 64 p^{5} T^{13} - 48 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 23 | \( 1 + 5 T - 71 T^{2} - 194 T^{3} + 4182 T^{4} + 6613 T^{5} - 145122 T^{6} - 44824 T^{7} + 4056959 T^{8} - 44824 p T^{9} - 145122 p^{2} T^{10} + 6613 p^{3} T^{11} + 4182 p^{4} T^{12} - 194 p^{5} T^{13} - 71 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 4 T + 75 T^{2} + 254 T^{3} + 2971 T^{4} + 254 p T^{5} + 75 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 17 T + 106 T^{2} + 371 T^{3} + 1762 T^{4} + 6322 T^{5} - 37335 T^{6} - 489212 T^{7} - 2888183 T^{8} - 489212 p T^{9} - 37335 p^{2} T^{10} + 6322 p^{3} T^{11} + 1762 p^{4} T^{12} + 371 p^{5} T^{13} + 106 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( 1 - 3 T - 118 T^{2} + 215 T^{3} + 8544 T^{4} - 8726 T^{5} - 446305 T^{6} + 133138 T^{7} + 18440725 T^{8} + 133138 p T^{9} - 446305 p^{2} T^{10} - 8726 p^{3} T^{11} + 8544 p^{4} T^{12} + 215 p^{5} T^{13} - 118 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 41 | \( ( 1 - 7 T + 117 T^{2} - 526 T^{3} + 6189 T^{4} - 526 p T^{5} + 117 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 15 T + 233 T^{2} + 1930 T^{3} + 373 p T^{4} + 1930 p T^{5} + 233 p^{2} T^{6} + 15 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 47 | \( 1 + 16 T + 78 T^{2} + 352 T^{3} + 2929 T^{4} - 384 T^{5} - 158674 T^{6} - 1962960 T^{7} - 17703580 T^{8} - 1962960 p T^{9} - 158674 p^{2} T^{10} - 384 p^{3} T^{11} + 2929 p^{4} T^{12} + 352 p^{5} T^{13} + 78 p^{6} T^{14} + 16 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( 1 + 4 T - 102 T^{2} - 658 T^{3} + 4123 T^{4} + 36287 T^{5} - 27469 T^{6} - 884999 T^{7} - 2291177 T^{8} - 884999 p T^{9} - 27469 p^{2} T^{10} + 36287 p^{3} T^{11} + 4123 p^{4} T^{12} - 658 p^{5} T^{13} - 102 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 59 | \( 1 + 17 T + 20 T^{2} - 799 T^{3} + 2880 T^{4} + 71162 T^{5} - 13995 T^{6} - 1871122 T^{7} - 3613343 T^{8} - 1871122 p T^{9} - 13995 p^{2} T^{10} + 71162 p^{3} T^{11} + 2880 p^{4} T^{12} - 799 p^{5} T^{13} + 20 p^{6} T^{14} + 17 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 - 9 T - 114 T^{2} + 1261 T^{3} + 7618 T^{4} - 83018 T^{5} - 385463 T^{6} + 1914956 T^{7} + 26133377 T^{8} + 1914956 p T^{9} - 385463 p^{2} T^{10} - 83018 p^{3} T^{11} + 7618 p^{4} T^{12} + 1261 p^{5} T^{13} - 114 p^{6} T^{14} - 9 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 5 T - 167 T^{2} + 690 T^{3} + 14502 T^{4} - 31305 T^{5} - 1373478 T^{6} + 366820 T^{7} + 116390827 T^{8} + 366820 p T^{9} - 1373478 p^{2} T^{10} - 31305 p^{3} T^{11} + 14502 p^{4} T^{12} + 690 p^{5} T^{13} - 167 p^{6} T^{14} - 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 - 6 T + 4 p T^{2} - 1235 T^{3} + 30199 T^{4} - 1235 p T^{5} + 4 p^{3} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( 1 + 15 T - 57 T^{2} - 740 T^{3} + 14212 T^{4} + 41065 T^{5} - 1446728 T^{6} - 4263550 T^{7} + 58157957 T^{8} - 4263550 p T^{9} - 1446728 p^{2} T^{10} + 41065 p^{3} T^{11} + 14212 p^{4} T^{12} - 740 p^{5} T^{13} - 57 p^{6} T^{14} + 15 p^{7} T^{15} + p^{8} T^{16} \) |
| 79 | \( 1 + 5 T - 2 T^{2} - 853 T^{3} - 11456 T^{4} - 44786 T^{5} + 123615 T^{6} + 5999536 T^{7} + 56254561 T^{8} + 5999536 p T^{9} + 123615 p^{2} T^{10} - 44786 p^{3} T^{11} - 11456 p^{4} T^{12} - 853 p^{5} T^{13} - 2 p^{6} T^{14} + 5 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 34 T + 724 T^{2} - 10255 T^{3} + 109139 T^{4} - 10255 p T^{5} + 724 p^{2} T^{6} - 34 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 3 T - 52 T^{2} + 433 T^{3} - 10352 T^{4} + 32694 T^{5} + 176901 T^{6} - 4270078 T^{7} + 84926465 T^{8} - 4270078 p T^{9} + 176901 p^{2} T^{10} + 32694 p^{3} T^{11} - 10352 p^{4} T^{12} + 433 p^{5} T^{13} - 52 p^{6} T^{14} - 3 p^{7} T^{15} + p^{8} T^{16} \) |
| 97 | \( ( 1 - 11 T + 332 T^{2} - 2687 T^{3} + 47041 T^{4} - 2687 p T^{5} + 332 p^{2} T^{6} - 11 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−4.94083162920627260103308850468, −4.83737809190836341822179158164, −4.62765486232109972069274737664, −4.45090106373387787698987366557, −4.35831630520152881800715406811, −4.22183833624505999635781442225, −3.74572359338933751464101907222, −3.73557216513676681985440906334, −3.69377054378696480846087269410, −3.62768166739564055533596790112, −3.34425747444229414350233196372, −3.33357764909304808064173435920, −3.33225699302734032263510027306, −3.30496312784933806741437118003, −2.77482128000148675221417492600, −2.63765395470309492304542278803, −2.52485907509119754069563183872, −2.33911418275809898540180512092, −1.87060408039596272752055400916, −1.80789410839218578470669896759, −1.79590891866650952783899250264, −1.58408852816267654747652885565, −1.43340147371719538051597970268, −0.78490294237490537732273058204, −0.55722097497934685351520986316,
0.55722097497934685351520986316, 0.78490294237490537732273058204, 1.43340147371719538051597970268, 1.58408852816267654747652885565, 1.79590891866650952783899250264, 1.80789410839218578470669896759, 1.87060408039596272752055400916, 2.33911418275809898540180512092, 2.52485907509119754069563183872, 2.63765395470309492304542278803, 2.77482128000148675221417492600, 3.30496312784933806741437118003, 3.33225699302734032263510027306, 3.33357764909304808064173435920, 3.34425747444229414350233196372, 3.62768166739564055533596790112, 3.69377054378696480846087269410, 3.73557216513676681985440906334, 3.74572359338933751464101907222, 4.22183833624505999635781442225, 4.35831630520152881800715406811, 4.45090106373387787698987366557, 4.62765486232109972069274737664, 4.83737809190836341822179158164, 4.94083162920627260103308850468
Plot not available for L-functions of degree greater than 10.
