| L(s) = 1 | − 4·4-s − 4·11-s + 10·16-s − 32·23-s + 20·25-s − 8·37-s + 16·44-s + 20·49-s + 56·53-s − 20·64-s − 24·67-s − 16·71-s + 128·92-s − 80·100-s + 24·113-s + 8·121-s + 127-s + 131-s + 137-s + 139-s + 32·148-s + 149-s + 151-s + 157-s + 163-s + 167-s − 84·169-s + ⋯ |
| L(s) = 1 | − 2·4-s − 1.20·11-s + 5/2·16-s − 6.67·23-s + 4·25-s − 1.31·37-s + 2.41·44-s + 20/7·49-s + 7.69·53-s − 5/2·64-s − 2.93·67-s − 1.89·71-s + 13.3·92-s − 8·100-s + 2.25·113-s + 8/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 2.63·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 6.46·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{16} \cdot 7^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3046119961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3046119961\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{4} \) |
| 3 | \( 1 \) |
| 7 | \( ( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | \( ( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| good | 5 | \( ( 1 - 2 p T^{2} + 54 T^{4} - 2 p^{3} T^{6} + p^{4} T^{8} )^{2} \) |
| 13 | \( ( 1 + 42 T^{2} + 758 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 + 20 T^{2} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 42 T^{2} + 974 T^{4} + 42 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 + 4 T + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 56 T^{2} + 1710 T^{4} - 56 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 48 T^{2} + 1154 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 2 T + 54 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 40 T^{2} + 2418 T^{4} + 40 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 43 | \( ( 1 + 4 T^{2} + 2358 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 47 | \( ( 1 - 64 T^{2} + 4098 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - 14 T + 134 T^{2} - 14 p T^{3} + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 - 226 T^{2} + 19710 T^{4} - 226 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 + 154 T^{2} + 11670 T^{4} + 154 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + 6 T + 122 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 + 2 T + p T^{2} )^{8} \) |
| 73 | \( ( 1 - 8 T^{2} - 1422 T^{4} - 8 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 79 | \( ( 1 - 142 T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 + 82 T^{2} + 13758 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{4} \) |
| 97 | \( ( 1 - 60 T^{2} + 11318 T^{4} - 60 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.10009880352403347777561178965, −4.08506533188069850925645417858, −3.92806769499810969378522327121, −3.92260474534398067938297659122, −3.62711239940977923416926413841, −3.40794353248715910125853897215, −3.28403455345688970922033534605, −3.21575577424505567524994307039, −3.03684669497185071045077912079, −2.94557002962312817736206494156, −2.80119288668105503570191762885, −2.53761138395734820327141860571, −2.42626054674739687636906035675, −2.23684250662594339649195938343, −2.14460513965739607571071282306, −2.13687339910709612899241854598, −1.92430273093018105522092504308, −1.83718892309898971538699177901, −1.37730285344933610776914106172, −1.16096228776303297672964214644, −1.03241790800917581745856716717, −0.969545393288141126297892176315, −0.69457561259195189409199766367, −0.27086454512546710710797861483, −0.13063235393916447917351413837,
0.13063235393916447917351413837, 0.27086454512546710710797861483, 0.69457561259195189409199766367, 0.969545393288141126297892176315, 1.03241790800917581745856716717, 1.16096228776303297672964214644, 1.37730285344933610776914106172, 1.83718892309898971538699177901, 1.92430273093018105522092504308, 2.13687339910709612899241854598, 2.14460513965739607571071282306, 2.23684250662594339649195938343, 2.42626054674739687636906035675, 2.53761138395734820327141860571, 2.80119288668105503570191762885, 2.94557002962312817736206494156, 3.03684669497185071045077912079, 3.21575577424505567524994307039, 3.28403455345688970922033534605, 3.40794353248715910125853897215, 3.62711239940977923416926413841, 3.92260474534398067938297659122, 3.92806769499810969378522327121, 4.08506533188069850925645417858, 4.10009880352403347777561178965
Plot not available for L-functions of degree greater than 10.
