| L(s) = 1 | + 8·2-s + 32·4-s + 96·8-s + 4·9-s + 264·16-s + 32·18-s + 672·32-s + 128·36-s + 1.53e3·64-s − 16·71-s + 384·72-s + 192·79-s + 8·81-s − 80·113-s + 127-s + 3.26e3·128-s + 131-s + 137-s + 139-s − 128·142-s + 1.05e3·144-s + 149-s + 151-s + 157-s + 1.53e3·158-s + 64·162-s + 163-s + ⋯ |
| L(s) = 1 | + 5.65·2-s + 16·4-s + 33.9·8-s + 4/3·9-s + 66·16-s + 7.54·18-s + 118.·32-s + 64/3·36-s + 192·64-s − 1.89·71-s + 45.2·72-s + 21.6·79-s + 8/9·81-s − 7.52·113-s + 0.0887·127-s + 288.·128-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 10.7·142-s + 88·144-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 122.·158-s + 5.02·162-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 7^{16} \cdot 13^{16}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{16} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(552.9005622\) |
| \(L(\frac12)\) |
\(\approx\) |
\(552.9005622\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{2} \) |
| 13 | \( 1 + 36 T^{2} + 648 T^{4} + 11160 T^{6} + 172319 T^{8} + 11160 p^{2} T^{10} + 648 p^{4} T^{12} + 36 p^{6} T^{14} + p^{8} T^{16} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 8 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 4 T^{2} + 8 T^{4} - 40 T^{6} - 161 T^{8} - 40 p^{2} T^{10} + 8 p^{4} T^{12} + 4 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 5 | \( ( 1 - 12 T^{2} + 72 T^{4} - 264 T^{6} + 959 T^{8} - 264 p^{2} T^{10} + 72 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )( 1 + 12 T^{2} + 72 T^{4} + 264 T^{6} + 959 T^{8} + 264 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 11 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
| 17 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 19 | \( ( 1 - 12 T^{2} + 72 T^{4} + 7800 T^{6} - 177121 T^{8} + 7800 p^{2} T^{10} + 72 p^{4} T^{12} - 12 p^{6} T^{14} + p^{8} T^{16} )( 1 + 12 T^{2} + 72 T^{4} - 7800 T^{6} - 177121 T^{8} - 7800 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 23 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{4}( 1 + 10 T^{2} - 429 T^{4} + 10 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{8} \) |
| 31 | \( ( 1 + p^{2} T^{4} )^{8} \) |
| 37 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
| 41 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
| 43 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{8} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{8} \) |
| 53 | \( ( 1 - p T^{2} )^{16} \) |
| 59 | \( ( 1 + 108 T^{2} + 5832 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 108 T^{2} + 5832 T^{4} - 122040 T^{6} - 18707521 T^{8} - 122040 p^{2} T^{10} + 5832 p^{4} T^{12} + 108 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 61 | \( ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 12 T^{2} + 72 T^{4} - 88440 T^{6} - 14376481 T^{8} - 88440 p^{2} T^{10} + 72 p^{4} T^{12} + 12 p^{6} T^{14} + p^{8} T^{16} ) \) |
| 67 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
| 71 | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{4}( 1 - 8 T + 32 T^{2} + 880 T^{3} - 8561 T^{4} + 880 p T^{5} + 32 p^{2} T^{6} - 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{8} \) |
| 79 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{8} \) |
| 83 | \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )^{2}( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
| 97 | \( ( 1 - p^{2} T^{4} + p^{4} T^{8} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{32} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−2.88056533539562337084728601849, −2.61978261613170898514140458136, −2.56211113361121454795042050129, −2.52866567168379219296217075559, −2.51707975235063092831241712790, −2.46104452567267054667948867935, −2.33933484939565047274110697579, −2.21699020202223391859774242989, −2.10554860057620952021689434805, −2.08255833024506090130748978502, −2.05085024351402051369105920173, −1.94562581726464004814105325326, −1.86922413020738212092636753981, −1.63206469494266028296720689634, −1.59435055305456697360215051448, −1.53851866819779757178067921584, −1.46542677698757557217243930343, −1.24450174246982871107456969768, −1.17733725138867153865003491862, −1.09888698547469099368664323669, −0.865167441555352699809887864993, −0.865069511884623711385372939182, −0.854616048722037483270136764408, −0.60669728597602715207507623405, −0.14668513627062788227937934034,
0.14668513627062788227937934034, 0.60669728597602715207507623405, 0.854616048722037483270136764408, 0.865069511884623711385372939182, 0.865167441555352699809887864993, 1.09888698547469099368664323669, 1.17733725138867153865003491862, 1.24450174246982871107456969768, 1.46542677698757557217243930343, 1.53851866819779757178067921584, 1.59435055305456697360215051448, 1.63206469494266028296720689634, 1.86922413020738212092636753981, 1.94562581726464004814105325326, 2.05085024351402051369105920173, 2.08255833024506090130748978502, 2.10554860057620952021689434805, 2.21699020202223391859774242989, 2.33933484939565047274110697579, 2.46104452567267054667948867935, 2.51707975235063092831241712790, 2.52866567168379219296217075559, 2.56211113361121454795042050129, 2.61978261613170898514140458136, 2.88056533539562337084728601849
Plot not available for L-functions of degree greater than 10.
