| L(s) = 1 | − 2·3-s + 3·9-s − 6·11-s + 4·17-s − 4·25-s + 4·27-s − 4·29-s + 4·31-s + 12·33-s − 12·37-s − 16·41-s + 26·49-s − 8·51-s + 12·67-s + 8·75-s − 11·81-s + 72·83-s + 8·87-s − 8·93-s − 8·97-s − 18·99-s + 52·101-s + 8·103-s + 24·111-s + 24·121-s + 32·123-s + 127-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 9-s − 1.80·11-s + 0.970·17-s − 4/5·25-s + 0.769·27-s − 0.742·29-s + 0.718·31-s + 2.08·33-s − 1.97·37-s − 2.49·41-s + 26/7·49-s − 1.12·51-s + 1.46·67-s + 0.923·75-s − 1.22·81-s + 7.90·83-s + 0.857·87-s − 0.829·93-s − 0.812·97-s − 1.80·99-s + 5.17·101-s + 0.788·103-s + 2.27·111-s + 2.18·121-s + 2.88·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8} \cdot 5^{8} \cdot 11^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.978008993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.978008993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 2 T + T^{2} - 8 T^{3} - 16 T^{4} - 8 p T^{5} + p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | \( ( 1 + T^{2} )^{4} \) |
| 11 | \( 1 + 6 T + 12 T^{2} - 6 p T^{3} - 378 T^{4} - 6 p^{2} T^{5} + 12 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| good | 7 | \( 1 - 26 T^{2} + 47 p T^{4} - 3098 T^{6} + 24180 T^{8} - 3098 p^{2} T^{10} + 47 p^{5} T^{12} - 26 p^{6} T^{14} + p^{8} T^{16} \) |
| 13 | \( 1 - 60 T^{2} + 1892 T^{4} - 39972 T^{6} + 607318 T^{8} - 39972 p^{2} T^{10} + 1892 p^{4} T^{12} - 60 p^{6} T^{14} + p^{8} T^{16} \) |
| 17 | \( ( 1 - 2 T + 37 T^{2} + 2 T^{3} + 36 p T^{4} + 2 p T^{5} + 37 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 19 | \( 1 - 54 T^{2} + 1513 T^{4} - 24406 T^{6} + 389156 T^{8} - 24406 p^{2} T^{10} + 1513 p^{4} T^{12} - 54 p^{6} T^{14} + p^{8} T^{16} \) |
| 23 | \( 1 - 80 T^{2} + 2588 T^{4} - 37552 T^{6} + 401926 T^{8} - 37552 p^{2} T^{10} + 2588 p^{4} T^{12} - 80 p^{6} T^{14} + p^{8} T^{16} \) |
| 29 | \( ( 1 + 2 T + 73 T^{2} + 124 T^{3} + 3004 T^{4} + 124 p T^{5} + 73 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 31 | \( ( 1 - 2 T + 9 T^{2} + 14 T^{3} + 636 T^{4} + 14 p T^{5} + 9 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 37 | \( ( 1 + 6 T + 77 T^{2} + 8 T^{3} + 1572 T^{4} + 8 p T^{5} + 77 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 41 | \( ( 1 + 8 T + 152 T^{2} + 904 T^{3} + 9166 T^{4} + 904 p T^{5} + 152 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 43 | \( 1 - 184 T^{2} + 16700 T^{4} - 1008840 T^{6} + 47686758 T^{8} - 1008840 p^{2} T^{10} + 16700 p^{4} T^{12} - 184 p^{6} T^{14} + p^{8} T^{16} \) |
| 47 | \( 1 - 120 T^{2} + 8828 T^{4} - 511176 T^{6} + 26922502 T^{8} - 511176 p^{2} T^{10} + 8828 p^{4} T^{12} - 120 p^{6} T^{14} + p^{8} T^{16} \) |
| 53 | \( 1 - 366 T^{2} + 60961 T^{4} - 6049838 T^{6} + 392003636 T^{8} - 6049838 p^{2} T^{10} + 60961 p^{4} T^{12} - 366 p^{6} T^{14} + p^{8} T^{16} \) |
| 59 | \( 1 - 284 T^{2} + 41876 T^{4} - 4064260 T^{6} + 281102518 T^{8} - 4064260 p^{2} T^{10} + 41876 p^{4} T^{12} - 284 p^{6} T^{14} + p^{8} T^{16} \) |
| 61 | \( 1 - 178 T^{2} + 16233 T^{4} - 1367798 T^{6} + 99891260 T^{8} - 1367798 p^{2} T^{10} + 16233 p^{4} T^{12} - 178 p^{6} T^{14} + p^{8} T^{16} \) |
| 67 | \( ( 1 - 6 T + 92 T^{2} - 270 T^{3} + 7174 T^{4} - 270 p T^{5} + 92 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 71 | \( 1 - 398 T^{2} + 73505 T^{4} - 8497986 T^{6} + 699860940 T^{8} - 8497986 p^{2} T^{10} + 73505 p^{4} T^{12} - 398 p^{6} T^{14} + p^{8} T^{16} \) |
| 73 | \( 1 - 204 T^{2} + 21444 T^{4} - 1995828 T^{6} + 165946934 T^{8} - 1995828 p^{2} T^{10} + 21444 p^{4} T^{12} - 204 p^{6} T^{14} + p^{8} T^{16} \) |
| 79 | \( 1 - 4 p T^{2} + 50260 T^{4} - 5841924 T^{6} + 529552214 T^{8} - 5841924 p^{2} T^{10} + 50260 p^{4} T^{12} - 4 p^{7} T^{14} + p^{8} T^{16} \) |
| 83 | \( ( 1 - 36 T + 724 T^{2} - 9924 T^{3} + 103462 T^{4} - 9924 p T^{5} + 724 p^{2} T^{6} - 36 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
| 89 | \( 1 - 450 T^{2} + 88625 T^{4} - 10694546 T^{6} + 1008754980 T^{8} - 10694546 p^{2} T^{10} + 88625 p^{4} T^{12} - 450 p^{6} T^{14} + p^{8} T^{16} \) |
| 97 | \( ( 1 + 4 T + 172 T^{2} + 732 T^{3} + 25958 T^{4} + 732 p T^{5} + 172 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} )^{2} \) |
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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
−3.63774262628700070316077480001, −3.45011738477178693693207448465, −3.35309982136982853390631045208, −3.31198775104654406891394977185, −3.27571456929968347869723990489, −3.19573958135364787287280103492, −3.06645209727711529753815696749, −2.98958180338778428871583008760, −2.76113577361030613726203250699, −2.42406490268166475222079481014, −2.27773480251639995818024650769, −2.24384590154903788338620633738, −2.15785693020796979252558298968, −2.07350992954652736834564294631, −2.01479665726920595455387385774, −1.90666043504541226691131731123, −1.78459550467572788329966557681, −1.32723689228536405378999956413, −1.24244507319561031508824570752, −1.04481745562654336350814567837, −0.876213955070272641785123011344, −0.78276280535055089158301410891, −0.54339682428973169266482312306, −0.49362768118819703381522321162, −0.14215217647355781052957450520,
0.14215217647355781052957450520, 0.49362768118819703381522321162, 0.54339682428973169266482312306, 0.78276280535055089158301410891, 0.876213955070272641785123011344, 1.04481745562654336350814567837, 1.24244507319561031508824570752, 1.32723689228536405378999956413, 1.78459550467572788329966557681, 1.90666043504541226691131731123, 2.01479665726920595455387385774, 2.07350992954652736834564294631, 2.15785693020796979252558298968, 2.24384590154903788338620633738, 2.27773480251639995818024650769, 2.42406490268166475222079481014, 2.76113577361030613726203250699, 2.98958180338778428871583008760, 3.06645209727711529753815696749, 3.19573958135364787287280103492, 3.27571456929968347869723990489, 3.31198775104654406891394977185, 3.35309982136982853390631045208, 3.45011738477178693693207448465, 3.63774262628700070316077480001
Plot not available for L-functions of degree greater than 10.
