| L(s) = 1 | + 24·23-s − 12·25-s + 32·71-s + 88·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + 233-s + 239-s + 241-s + ⋯ |
| L(s) = 1 | + 5.00·23-s − 2.39·25-s + 3.79·71-s + 8·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + 0.0663·227-s + 0.0660·229-s + 0.0655·233-s + 0.0646·239-s + 0.0644·241-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{40} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.406196968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.406196968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 12 T^{2} + 72 T^{4} + 12 p^{2} T^{6} + p^{4} T^{8} \) |
| 7 | \( ( 1 + p^{2} T^{4} )^{2} \) |
| good | 3 | \( ( 1 - 4 T^{2} + 8 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} )( 1 + 4 T^{2} + 8 T^{4} + 4 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 11 | \( ( 1 - p T^{2} )^{8} \) |
| 13 | \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 17 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 - 10 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | \( ( 1 + p T^{2} )^{8} \) |
| 31 | \( ( 1 - p T^{2} )^{8} \) |
| 37 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 47 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 59 | \( ( 1 + 108 T^{2} + 5832 T^{4} + 108 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 61 | \( ( 1 - 12 T^{2} + 72 T^{4} - 12 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 67 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 71 | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{4} \) |
| 73 | \( ( 1 + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - 24 T + 288 T^{2} - 24 p T^{3} + p^{2} T^{4} )^{2}( 1 + 24 T + 288 T^{2} + 24 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 - 36 T^{2} + 648 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} )( 1 + 36 T^{2} + 648 T^{4} + 36 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 89 | \( ( 1 + p T^{2} )^{8} \) |
| 97 | \( ( 1 + p^{2} T^{4} )^{4} \) |
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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
−4.19289724940836109722008331196, −4.15559894726787507964281741132, −3.70802921825022489327391669438, −3.69318461632874718489411016173, −3.65599229508767578457780372132, −3.59500492625356168714462192381, −3.55225653211854200401294995374, −3.29060467393481638455196348415, −3.06170289917278022764448738124, −3.04581784440595423325050525473, −2.87636288762870517391529274895, −2.83928905534890245111735938650, −2.52703751834728211075253535942, −2.41458152777243426972978050309, −2.19227867426395781809993512840, −2.07944165355467800456103417739, −1.96734492080600686819250883195, −1.91854399737189503265027566937, −1.54182148848421050523365971561, −1.33923628580474456271374830985, −1.21621844897670980363648807818, −0.945198921250729642269508848638, −0.68279097708874942884705672610, −0.63894087885402324468949746112, −0.34606467007836696125367454094,
0.34606467007836696125367454094, 0.63894087885402324468949746112, 0.68279097708874942884705672610, 0.945198921250729642269508848638, 1.21621844897670980363648807818, 1.33923628580474456271374830985, 1.54182148848421050523365971561, 1.91854399737189503265027566937, 1.96734492080600686819250883195, 2.07944165355467800456103417739, 2.19227867426395781809993512840, 2.41458152777243426972978050309, 2.52703751834728211075253535942, 2.83928905534890245111735938650, 2.87636288762870517391529274895, 3.04581784440595423325050525473, 3.06170289917278022764448738124, 3.29060467393481638455196348415, 3.55225653211854200401294995374, 3.59500492625356168714462192381, 3.65599229508767578457780372132, 3.69318461632874718489411016173, 3.70802921825022489327391669438, 4.15559894726787507964281741132, 4.19289724940836109722008331196
Plot not available for L-functions of degree greater than 10.
