| 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 − 21.3·36-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
| \(L(1)\) |
\(\approx\) |
\(0.02703802446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02703802446\) |
| \(L(\frac{3}{2})\) |
|
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
−2.71836531834098202005329723779, −2.67421396722663073100672536086, −2.60968123829473431152909448280, −2.52087730527314128946957976178, −2.51378736494528842433819439276, −2.33965303724404413388445460091, −2.19637690766803017755701670651, −2.17127448941241412634470675879, −2.11799542841338198229186388407, −1.79467098541729424316744075302, −1.69850954734213929687556131329, −1.53729074663556180885445615894, −1.49635076727650602350991668897, −1.45341881383639260930659190888, −1.35798945278540687277917260104, −1.35442331619770142722822551241, −1.28125056705659874287775770418, −1.15558051332144011479313603234, −1.11229364334182168439007777725, −0.880839735395319041732578724811, −0.43609385515291496791811942954, −0.42519192493479483170907744846, −0.31789703509482493269608046225, −0.26517568709651692271876127707, −0.21367467882191829416472010603,
0.21367467882191829416472010603, 0.26517568709651692271876127707, 0.31789703509482493269608046225, 0.42519192493479483170907744846, 0.43609385515291496791811942954, 0.880839735395319041732578724811, 1.11229364334182168439007777725, 1.15558051332144011479313603234, 1.28125056705659874287775770418, 1.35442331619770142722822551241, 1.35798945278540687277917260104, 1.45341881383639260930659190888, 1.49635076727650602350991668897, 1.53729074663556180885445615894, 1.69850954734213929687556131329, 1.79467098541729424316744075302, 2.11799542841338198229186388407, 2.17127448941241412634470675879, 2.19637690766803017755701670651, 2.33965303724404413388445460091, 2.51378736494528842433819439276, 2.52087730527314128946957976178, 2.60968123829473431152909448280, 2.67421396722663073100672536086, 2.71836531834098202005329723779
Plot not available for L-functions of degree greater than 10.
