| L(s) = 1 | + 2·5-s − 4·11-s − 4·19-s + 25-s + 4·29-s − 4·31-s − 28·41-s − 3·49-s − 8·55-s + 16·59-s + 4·61-s + 12·71-s + 40·79-s + 4·89-s − 8·95-s − 60·101-s + 20·109-s − 14·121-s + 8·125-s + 127-s + 131-s + 137-s + 139-s + 8·145-s + 149-s + 151-s − 8·155-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 1.20·11-s − 0.917·19-s + 1/5·25-s + 0.742·29-s − 0.718·31-s − 4.37·41-s − 3/7·49-s − 1.07·55-s + 2.08·59-s + 0.512·61-s + 1.42·71-s + 4.50·79-s + 0.423·89-s − 0.820·95-s − 5.97·101-s + 1.91·109-s − 1.27·121-s + 0.715·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.664·145-s + 0.0819·149-s + 0.0813·151-s − 0.642·155-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 3^{12} \cdot 5^{6} \cdot 7^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8226565384\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8226565384\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} \) |
| 7 | \( ( 1 + T^{2} )^{3} \) |
| good | 11 | \( ( 1 + 2 T + 13 T^{2} + 36 T^{3} + 13 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 13 | \( ( 1 - 8 T + 7 T^{2} + 64 T^{3} + 7 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )( 1 + 8 T + 7 T^{2} - 64 T^{3} + 7 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 17 | \( 1 - 70 T^{2} + 2415 T^{4} - 51220 T^{6} + 2415 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 + 2 T + 45 T^{2} + 68 T^{3} + 45 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 90 T^{2} + 3839 T^{4} - 105452 T^{6} + 3839 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 - 2 T + 3 T^{2} - 12 T^{3} + 3 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 2 T + 81 T^{2} + 116 T^{3} + 81 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( ( 1 - 10 T - 5 T^{2} + 356 T^{3} - 5 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )( 1 + 10 T - 5 T^{2} - 356 T^{3} - 5 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} ) \) |
| 41 | \( ( 1 + 14 T + 175 T^{2} + 1188 T^{3} + 175 p T^{4} + 14 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 43 | \( 1 - 82 T^{2} + 5399 T^{4} - 217692 T^{6} + 5399 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 90 T^{2} + 5231 T^{4} - 236204 T^{6} + 5231 p^{2} T^{8} - 90 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 66 T^{2} + 2711 T^{4} + 8452 T^{6} + 2711 p^{2} T^{8} - 66 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 - 8 T + 113 T^{2} - 688 T^{3} + 113 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 - 2 T + 35 T^{2} - 780 T^{3} + 35 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 67 | \( ( 1 - p T^{2} )^{6} \) |
| 71 | \( ( 1 - 6 T + 113 T^{2} - 1052 T^{3} + 113 p T^{4} - 6 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 154 T^{2} + 21503 T^{4} - 1698540 T^{6} + 21503 p^{2} T^{8} - 154 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 - 20 T + 317 T^{2} - 3096 T^{3} + 317 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 306 T^{2} + 47783 T^{4} - 4793948 T^{6} + 47783 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 - 2 T + 207 T^{2} - 156 T^{3} + 207 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 394 T^{2} + 77839 T^{4} - 9393484 T^{6} + 77839 p^{2} T^{8} - 394 p^{4} T^{10} + p^{6} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.68581386536221207652273383825, −4.64396588908693837579906487429, −4.16343818137939350829768178121, −4.08723990834652325301504129882, −4.05532566677377293450828484909, −4.05350306090576101591252575665, −3.55592530133416368806222763975, −3.52452300524485334027501002105, −3.43613842668041756987824931893, −3.35770122588124166342951182956, −3.10799741175284006871333470341, −2.90864718630323610424312967692, −2.58011024298712193314680025982, −2.46979072961273578477879157300, −2.46544849542397634251459926215, −2.42363650973790615175832615393, −1.92787604463842907377696639753, −1.78946046660152835524318613624, −1.77049294945853635274867955725, −1.62319907664846967585794287814, −1.39044425112086682275302086815, −0.934014657597847936100595320934, −0.64126499306672039365435471385, −0.61722806396296162713236386538, −0.10272849062823947252017543617,
0.10272849062823947252017543617, 0.61722806396296162713236386538, 0.64126499306672039365435471385, 0.934014657597847936100595320934, 1.39044425112086682275302086815, 1.62319907664846967585794287814, 1.77049294945853635274867955725, 1.78946046660152835524318613624, 1.92787604463842907377696639753, 2.42363650973790615175832615393, 2.46544849542397634251459926215, 2.46979072961273578477879157300, 2.58011024298712193314680025982, 2.90864718630323610424312967692, 3.10799741175284006871333470341, 3.35770122588124166342951182956, 3.43613842668041756987824931893, 3.52452300524485334027501002105, 3.55592530133416368806222763975, 4.05350306090576101591252575665, 4.05532566677377293450828484909, 4.08723990834652325301504129882, 4.16343818137939350829768178121, 4.64396588908693837579906487429, 4.68581386536221207652273383825
Plot not available for L-functions of degree greater than 10.
