| L(s) = 1 | − 3·9-s + 8·19-s + 25-s − 8·31-s + 12·41-s + 10·49-s − 32·59-s − 24·61-s − 16·71-s − 40·79-s + 6·81-s − 20·89-s − 16·101-s + 24·109-s − 34·121-s + 16·125-s + ⋯ |
| L(s) = 1 | − 9-s + 1.83·19-s + 1/5·25-s − 1.43·31-s + 1.87·41-s + 10/7·49-s − 4.16·59-s − 3.07·61-s − 1.89·71-s − 4.50·79-s + 2/3·81-s − 2.11·89-s − 1.59·101-s + 2.29·109-s − 3.09·121-s + 1.43·125-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{6} \cdot 5^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.265854446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.265854446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( ( 1 + T^{2} )^{3} \) |
| 5 | \( 1 - T^{2} - 16 T^{3} - p T^{4} + p^{3} T^{6} \) |
| good | 7 | \( 1 - 10 T^{2} + 31 T^{4} + 116 T^{6} + 31 p^{2} T^{8} - 10 p^{4} T^{10} + p^{6} T^{12} \) |
| 11 | \( ( 1 + 17 T^{2} + 16 T^{3} + 17 p T^{4} + p^{3} T^{6} )^{2} \) |
| 13 | \( 1 - 34 T^{2} + 743 T^{4} - 10620 T^{6} + 743 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} \) |
| 17 | \( 1 - 70 T^{2} + 2351 T^{4} - 49044 T^{6} + 2351 p^{2} T^{8} - 70 p^{4} T^{10} + p^{6} T^{12} \) |
| 19 | \( ( 1 - 4 T + 41 T^{2} - 120 T^{3} + 41 p T^{4} - 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 23 | \( 1 - 74 T^{2} + 2559 T^{4} - 62732 T^{6} + 2559 p^{2} T^{8} - 74 p^{4} T^{10} + p^{6} T^{12} \) |
| 29 | \( ( 1 + 71 T^{2} - 16 T^{3} + 71 p T^{4} + p^{3} T^{6} )^{2} \) |
| 31 | \( ( 1 + 4 T + 61 T^{2} + 280 T^{3} + 61 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 37 | \( 1 - 50 T^{2} + 3511 T^{4} - 3692 p T^{6} + 3511 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} \) |
| 41 | \( ( 1 - 2 T + p T^{2} )^{6} \) |
| 43 | \( 1 - 82 T^{2} + 4375 T^{4} - 227932 T^{6} + 4375 p^{2} T^{8} - 82 p^{4} T^{10} + p^{6} T^{12} \) |
| 47 | \( 1 - 186 T^{2} + 17903 T^{4} - 1043180 T^{6} + 17903 p^{2} T^{8} - 186 p^{4} T^{10} + p^{6} T^{12} \) |
| 53 | \( 1 - 162 T^{2} + 10007 T^{4} - 448316 T^{6} + 10007 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} \) |
| 59 | \( ( 1 + 16 T + 3 p T^{2} + 1296 T^{3} + 3 p^{2} T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 61 | \( ( 1 + 4 T + p T^{2} )^{6} \) |
| 67 | \( 1 - 98 T^{2} + 13255 T^{4} - 805628 T^{6} + 13255 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} \) |
| 71 | \( ( 1 + 8 T + 149 T^{2} + 880 T^{3} + 149 p T^{4} + 8 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 73 | \( 1 - 246 T^{2} + 28991 T^{4} - 2372468 T^{6} + 28991 p^{2} T^{8} - 246 p^{4} T^{10} + p^{6} T^{12} \) |
| 79 | \( ( 1 + 20 T + 3 p T^{2} + 1976 T^{3} + 3 p^{2} T^{4} + 20 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 83 | \( 1 - 194 T^{2} + 29799 T^{4} - 2725052 T^{6} + 29799 p^{2} T^{8} - 194 p^{4} T^{10} + p^{6} T^{12} \) |
| 89 | \( ( 1 + 10 T + 87 T^{2} + 12 p T^{3} + 87 p T^{4} + 10 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| 97 | \( 1 - 390 T^{2} + 71759 T^{4} - 8391188 T^{6} + 71759 p^{2} T^{8} - 390 p^{4} T^{10} + p^{6} T^{12} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.28912884876736660813795918175, −4.23855473708064684037861623169, −4.15820204280121641504390250131, −4.05883908637779616511534236383, −3.80833603172745733719157381957, −3.53030262641492913512612142945, −3.45186472414289885571694061561, −3.29553725449462646769857004544, −3.23629937342424649957006969629, −3.01007830854384238521773490571, −2.90605486321784005316444008527, −2.71742402388887386116869418428, −2.61172320359741213200102527064, −2.54685888328409086185589039175, −2.42378280514083631598095436196, −2.14248899572517487226869210709, −1.69089712591479485681301206596, −1.60143233000083190697664076171, −1.59499843572966612599732450932, −1.26539455290586705993835873472, −1.21961200746891784332335780592, −1.20286608484545189500234690108, −0.58974749725239092460303661280, −0.25842170283083566375346572672, −0.20648577479924331264933810837,
0.20648577479924331264933810837, 0.25842170283083566375346572672, 0.58974749725239092460303661280, 1.20286608484545189500234690108, 1.21961200746891784332335780592, 1.26539455290586705993835873472, 1.59499843572966612599732450932, 1.60143233000083190697664076171, 1.69089712591479485681301206596, 2.14248899572517487226869210709, 2.42378280514083631598095436196, 2.54685888328409086185589039175, 2.61172320359741213200102527064, 2.71742402388887386116869418428, 2.90605486321784005316444008527, 3.01007830854384238521773490571, 3.23629937342424649957006969629, 3.29553725449462646769857004544, 3.45186472414289885571694061561, 3.53030262641492913512612142945, 3.80833603172745733719157381957, 4.05883908637779616511534236383, 4.15820204280121641504390250131, 4.23855473708064684037861623169, 4.28912884876736660813795918175
Plot not available for L-functions of degree greater than 10.
