| L(s) = 1 | − 6·5-s + 2·13-s + 5·25-s + 6·29-s + 8·37-s − 24·41-s − 19·49-s + 2·61-s − 12·65-s + 4·73-s + 4·97-s + 66·101-s − 16·109-s − 30·113-s + 32·121-s + 42·125-s + 127-s + 131-s + 137-s + 139-s − 36·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 37·169-s + ⋯ |
| L(s) = 1 | − 2.68·5-s + 0.554·13-s + 25-s + 1.11·29-s + 1.31·37-s − 3.74·41-s − 2.71·49-s + 0.256·61-s − 1.48·65-s + 0.468·73-s + 0.406·97-s + 6.56·101-s − 1.53·109-s − 2.82·113-s + 2.90·121-s + 3.75·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 2.98·145-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.84·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5661829925\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5661829925\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( ( 1 + 3 T + 11 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 11 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 7 | \( 1 + 19 T^{2} + 181 T^{4} + 1558 T^{6} + 12310 T^{8} + 1558 p^{2} T^{10} + 181 p^{4} T^{12} + 19 p^{6} T^{14} + p^{8} T^{16} \) |
| 11 | \( 1 - 32 T^{2} + 559 T^{4} - 7136 T^{6} + 77680 T^{8} - 7136 p^{2} T^{10} + 559 p^{4} T^{12} - 32 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( ( 1 - T - 17 T^{2} + 8 T^{3} + 142 T^{4} + 8 p T^{5} - 17 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 61 T^{2} + 1500 T^{4} - 61 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 - 49 T^{2} + 1248 T^{4} - 49 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( 1 - 77 T^{2} + 3397 T^{4} - 113498 T^{6} + 2994742 T^{8} - 113498 p^{2} T^{10} + 3397 p^{4} T^{12} - 77 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 - 3 T + 59 T^{2} - 168 T^{3} + 2382 T^{4} - 168 p T^{5} + 59 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( 1 + 55 T^{2} + 1345 T^{4} - 13310 T^{6} - 1008146 T^{8} - 13310 p^{2} T^{10} + 1345 p^{4} T^{12} + 55 p^{6} T^{14} + p^{8} T^{16} \) |
| 37 | \( ( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 + 12 T + 131 T^{2} + 996 T^{3} + 7176 T^{4} + 996 p T^{5} + 131 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 + 64 T^{2} - 593 T^{4} + 63424 T^{6} + 10994416 T^{8} + 63424 p^{2} T^{10} - 593 p^{4} T^{12} + 64 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 53 T^{2} + 2053 T^{4} + 194086 T^{6} - 10513226 T^{8} + 194086 p^{2} T^{10} + 2053 p^{4} T^{12} - 53 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( ( 1 - 136 T^{2} + 9054 T^{4} - 136 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 59 | \( 1 - 56 T^{2} - 617 T^{4} + 179704 T^{6} - 8948768 T^{8} + 179704 p^{2} T^{10} - 617 p^{4} T^{12} - 56 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( ( 1 - T - 113 T^{2} + 8 T^{3} + 9214 T^{4} + 8 p T^{5} - 113 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 67 | \( 1 + 160 T^{2} + 10255 T^{4} + 1018720 T^{6} + 100399504 T^{8} + 1018720 p^{2} T^{10} + 10255 p^{4} T^{12} + 160 p^{6} T^{14} + p^{8} T^{16} \) |
| 71 | \( ( 1 + 140 T^{2} + 10230 T^{4} + 140 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 73 | \( ( 1 - T + 138 T^{2} - p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( 1 + 115 T^{2} - 2555 T^{4} + 379270 T^{6} + 127521094 T^{8} + 379270 p^{2} T^{10} - 2555 p^{4} T^{12} + 115 p^{6} T^{14} + p^{8} T^{16} \) |
| 83 | \( 1 - 221 T^{2} + 22861 T^{4} - 2696642 T^{6} + 291036430 T^{8} - 2696642 p^{2} T^{10} + 22861 p^{4} T^{12} - 221 p^{6} T^{14} + p^{8} T^{16} \) |
| 89 | \( ( 1 - 184 T^{2} + 21006 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 97 | \( ( 1 - 2 T - 59 T^{2} + 262 T^{3} - 5828 T^{4} + 262 p T^{5} - 59 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.88074580891553822842346961778, −3.74098737810386314294192071421, −3.60771864916944807005549439080, −3.59059415980102795349685980664, −3.49342858486640232749479682849, −3.32742528744028794654925122170, −3.23905772667244709055868097554, −3.17131780330014694596399757639, −3.02408282208190283879432884505, −2.90677093891733064306820097722, −2.66786393125838756279043477595, −2.54911171869896736911343563463, −2.34720720293436557848111278739, −2.26608696023373612616145897765, −2.02905897925084089394173579564, −1.86230638476041415293179096631, −1.68099078736303798302124313777, −1.55765566820182753069189586846, −1.51178308547729397445461668758, −1.48589992828108610432835399879, −0.77737210546552802990792038105, −0.69708161300708096054089383316, −0.69339152863933710561095272778, −0.46203918409998538061410207575, −0.10876514460939150405045194650,
0.10876514460939150405045194650, 0.46203918409998538061410207575, 0.69339152863933710561095272778, 0.69708161300708096054089383316, 0.77737210546552802990792038105, 1.48589992828108610432835399879, 1.51178308547729397445461668758, 1.55765566820182753069189586846, 1.68099078736303798302124313777, 1.86230638476041415293179096631, 2.02905897925084089394173579564, 2.26608696023373612616145897765, 2.34720720293436557848111278739, 2.54911171869896736911343563463, 2.66786393125838756279043477595, 2.90677093891733064306820097722, 3.02408282208190283879432884505, 3.17131780330014694596399757639, 3.23905772667244709055868097554, 3.32742528744028794654925122170, 3.49342858486640232749479682849, 3.59059415980102795349685980664, 3.60771864916944807005549439080, 3.74098737810386314294192071421, 3.88074580891553822842346961778
Plot not available for L-functions of degree greater than 10.
