| L(s) = 1 | + 4·5-s + 7-s + 11-s − 4·13-s − 10·17-s + 2·19-s + 7·23-s + 6·25-s + 7·29-s + 2·31-s + 4·35-s + 12·37-s + 12·41-s + 11·43-s + 7·47-s + 13·49-s − 24·53-s + 4·55-s + 11·59-s − 19·61-s − 16·65-s + 10·67-s − 24·71-s + 18·73-s + 77-s + 24·79-s + 23·83-s + ⋯ |
| L(s) = 1 | + 1.78·5-s + 0.377·7-s + 0.301·11-s − 1.10·13-s − 2.42·17-s + 0.458·19-s + 1.45·23-s + 6/5·25-s + 1.29·29-s + 0.359·31-s + 0.676·35-s + 1.97·37-s + 1.87·41-s + 1.67·43-s + 1.02·47-s + 13/7·49-s − 3.29·53-s + 0.539·55-s + 1.43·59-s − 2.43·61-s − 1.98·65-s + 1.22·67-s − 2.84·71-s + 2.10·73-s + 0.113·77-s + 2.70·79-s + 2.52·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{24} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.004812829\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.004812829\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( ( 1 - T + T^{2} )^{4} \) |
| good | 7 | \( ( 1 - 2 T + 4 T^{2} + 10 T^{3} - 41 T^{4} + 10 p T^{5} + 4 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )( 1 + T - 2 p T^{2} + T^{3} + 142 T^{4} + p T^{5} - 2 p^{3} T^{6} + p^{3} T^{7} + p^{4} T^{8} ) \) |
| 11 | \( 1 - T - T^{2} + 24 T^{3} - 247 T^{4} + 307 T^{5} + 34 T^{6} - 4095 T^{7} + 42322 T^{8} - 4095 p T^{9} + 34 p^{2} T^{10} + 307 p^{3} T^{11} - 247 p^{4} T^{12} + 24 p^{5} T^{13} - p^{6} T^{14} - p^{7} T^{15} + p^{8} T^{16} \) |
| 13 | \( 1 + 4 T - 12 T^{2} - 160 T^{3} - 166 T^{4} + 2328 T^{5} + 9520 T^{6} - 16076 T^{7} - 166365 T^{8} - 16076 p T^{9} + 9520 p^{2} T^{10} + 2328 p^{3} T^{11} - 166 p^{4} T^{12} - 160 p^{5} T^{13} - 12 p^{6} T^{14} + 4 p^{7} T^{15} + p^{8} T^{16} \) |
| 17 | \( ( 1 + 5 T + 38 T^{2} + 215 T^{3} + 886 T^{4} + 215 p T^{5} + 38 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - T - 8 T^{2} - 25 T^{3} + 322 T^{4} - 25 p T^{5} - 8 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 7 T - 2 p T^{2} + 9 p T^{3} + 2723 T^{4} - 5948 T^{5} - 88340 T^{6} + 51366 T^{7} + 2328592 T^{8} + 51366 p T^{9} - 88340 p^{2} T^{10} - 5948 p^{3} T^{11} + 2723 p^{4} T^{12} + 9 p^{6} T^{13} - 2 p^{7} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 29 | \( 1 - 7 T - 52 T^{2} + 201 T^{3} + 3053 T^{4} - 3440 T^{5} - 117134 T^{6} + 91842 T^{7} + 2887336 T^{8} + 91842 p T^{9} - 117134 p^{2} T^{10} - 3440 p^{3} T^{11} + 3053 p^{4} T^{12} + 201 p^{5} T^{13} - 52 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 31 | \( 1 - 2 T - 60 T^{2} + 284 T^{3} + 1214 T^{4} - 9690 T^{5} - 5408 T^{6} + 125998 T^{7} + 163935 T^{8} + 125998 p T^{9} - 5408 p^{2} T^{10} - 9690 p^{3} T^{11} + 1214 p^{4} T^{12} + 284 p^{5} T^{13} - 60 p^{6} T^{14} - 2 p^{7} T^{15} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 6 T + 100 T^{2} - 414 T^{3} + 4806 T^{4} - 414 p T^{5} + 100 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( 1 - 12 T + 10 T^{2} - 240 T^{3} + 4489 T^{4} + 6516 T^{5} - 122870 T^{6} + 388536 T^{7} - 4685564 T^{8} + 388536 p T^{9} - 122870 p^{2} T^{10} + 6516 p^{3} T^{11} + 4489 p^{4} T^{12} - 240 p^{5} T^{13} + 10 p^{6} T^{14} - 12 p^{7} T^{15} + p^{8} T^{16} \) |
| 43 | \( 1 - 11 T - 57 T^{2} + 512 T^{3} + 6701 T^{4} - 22179 T^{5} - 429530 T^{6} + 333931 T^{7} + 21567906 T^{8} + 333931 p T^{9} - 429530 p^{2} T^{10} - 22179 p^{3} T^{11} + 6701 p^{4} T^{12} + 512 p^{5} T^{13} - 57 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 47 | \( 1 - 7 T - 34 T^{2} + 111 T^{3} + 11 T^{4} + 17404 T^{5} + 2728 T^{6} - 852474 T^{7} + 4306456 T^{8} - 852474 p T^{9} + 2728 p^{2} T^{10} + 17404 p^{3} T^{11} + 11 p^{4} T^{12} + 111 p^{5} T^{13} - 34 p^{6} T^{14} - 7 p^{7} T^{15} + p^{8} T^{16} \) |
| 53 | \( ( 1 + 12 T + 164 T^{2} + 1620 T^{3} + 11910 T^{4} + 1620 p T^{5} + 164 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 11 T - 49 T^{2} + 864 T^{3} + 293 T^{4} - 14503 T^{5} - 141854 T^{6} + 1749 p T^{7} + 184334 p T^{8} + 1749 p^{2} T^{9} - 141854 p^{2} T^{10} - 14503 p^{3} T^{11} + 293 p^{4} T^{12} + 864 p^{5} T^{13} - 49 p^{6} T^{14} - 11 p^{7} T^{15} + p^{8} T^{16} \) |
| 61 | \( 1 + 19 T + 24 T^{2} - 601 T^{3} + 13373 T^{4} + 114792 T^{5} - 629606 T^{6} + 2027290 T^{7} + 112200912 T^{8} + 2027290 p T^{9} - 629606 p^{2} T^{10} + 114792 p^{3} T^{11} + 13373 p^{4} T^{12} - 601 p^{5} T^{13} + 24 p^{6} T^{14} + 19 p^{7} T^{15} + p^{8} T^{16} \) |
| 67 | \( 1 - 10 T - 108 T^{2} + 1180 T^{3} + 6617 T^{4} - 41310 T^{5} - 799952 T^{6} - 79000 T^{7} + 81777492 T^{8} - 79000 p T^{9} - 799952 p^{2} T^{10} - 41310 p^{3} T^{11} + 6617 p^{4} T^{12} + 1180 p^{5} T^{13} - 108 p^{6} T^{14} - 10 p^{7} T^{15} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 12 T + 308 T^{2} + 2520 T^{3} + 33582 T^{4} + 2520 p T^{5} + 308 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - 9 T + 244 T^{2} - 1395 T^{3} + 23814 T^{4} - 1395 p T^{5} + 244 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 79 | \( 1 - 24 T + 380 T^{2} - 6000 T^{3} + 73354 T^{4} - 752328 T^{5} + 7894640 T^{6} - 73819992 T^{7} + 631921171 T^{8} - 73819992 p T^{9} + 7894640 p^{2} T^{10} - 752328 p^{3} T^{11} + 73354 p^{4} T^{12} - 6000 p^{5} T^{13} + 380 p^{6} T^{14} - 24 p^{7} T^{15} + p^{8} T^{16} \) |
| 83 | \( 1 - 23 T + 104 T^{2} - 135 T^{3} + 18269 T^{4} - 79744 T^{5} - 1999130 T^{6} + 11349204 T^{7} + 36715516 T^{8} + 11349204 p T^{9} - 1999130 p^{2} T^{10} - 79744 p^{3} T^{11} + 18269 p^{4} T^{12} - 135 p^{5} T^{13} + 104 p^{6} T^{14} - 23 p^{7} T^{15} + p^{8} T^{16} \) |
| 89 | \( ( 1 + 21 T + 329 T^{2} + 3366 T^{3} + 35214 T^{4} + 3366 p T^{5} + 329 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 97 | \( 1 + T - 177 T^{2} - 2860 T^{3} + 15659 T^{4} + 388713 T^{5} + 2927554 T^{6} - 27832019 T^{7} - 398208672 T^{8} - 27832019 p T^{9} + 2927554 p^{2} T^{10} + 388713 p^{3} T^{11} + 15659 p^{4} T^{12} - 2860 p^{5} T^{13} - 177 p^{6} T^{14} + p^{7} T^{15} + p^{8} T^{16} \) |
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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
−4.37069713461188862589829471794, −4.26227224941482504106606924781, −4.22400705082640872308672590010, −3.85922946104022314147471989910, −3.65098772228256461860620859824, −3.43301069633804439073582893061, −3.40635590676707286282630290763, −3.33120465036180244806466592306, −3.12130996607736251530381352828, −2.98480513845315864950342964793, −2.77169978114876326429436014321, −2.53914591898204189310624212829, −2.49775448509002805026077136166, −2.39023552551287810274917947366, −2.37483150202727251312982349793, −2.20460495403976045166222154837, −2.00615219801429480592100265342, −1.97141927535810777276481625823, −1.56511647287137743928577485972, −1.34894203617498501371268379662, −1.16869806871025966475438830739, −1.13135937835222281379359830151, −0.807043190675535263428212814719, −0.64857077270617187724019591178, −0.24687139960939972495027519170,
0.24687139960939972495027519170, 0.64857077270617187724019591178, 0.807043190675535263428212814719, 1.13135937835222281379359830151, 1.16869806871025966475438830739, 1.34894203617498501371268379662, 1.56511647287137743928577485972, 1.97141927535810777276481625823, 2.00615219801429480592100265342, 2.20460495403976045166222154837, 2.37483150202727251312982349793, 2.39023552551287810274917947366, 2.49775448509002805026077136166, 2.53914591898204189310624212829, 2.77169978114876326429436014321, 2.98480513845315864950342964793, 3.12130996607736251530381352828, 3.33120465036180244806466592306, 3.40635590676707286282630290763, 3.43301069633804439073582893061, 3.65098772228256461860620859824, 3.85922946104022314147471989910, 4.22400705082640872308672590010, 4.26227224941482504106606924781, 4.37069713461188862589829471794
Plot not available for L-functions of degree greater than 10.
