| L(s) = 1 | − 24·17-s − 20·25-s + 12·41-s + 28·49-s − 8·73-s − 144·89-s + 20·97-s + 72·113-s − 14·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 52·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + ⋯ |
| L(s) = 1 | − 5.82·17-s − 4·25-s + 1.87·41-s + 4·49-s − 0.936·73-s − 15.2·89-s + 2.03·97-s + 6.77·113-s − 1.27·121-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 − 4·169-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{48} \cdot 3^{24}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5558238747\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5558238747\) |
| \(L(\frac{3}{2})\) |
|
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 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 7 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 11 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}( 1 - 14 T^{2} + 75 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 13 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 17 | \( ( 1 + 6 T + 19 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{4} \) |
| 19 | \( ( 1 + 34 T^{2} + 795 T^{4} + 34 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 23 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 29 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - 6 T + p T^{2} )^{4}( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2}( 1 - 14 T^{2} - 1653 T^{4} - 14 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 47 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2}( 1 + 82 T^{2} + 3243 T^{4} + 82 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 61 | \( ( 1 + p T^{2} + p^{2} T^{4} )^{4} \) |
| 67 | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2}( 1 - 62 T^{2} - 645 T^{4} - 62 p^{2} T^{6} + p^{4} T^{8} ) \) |
| 71 | \( ( 1 + p T^{2} )^{8} \) |
| 73 | \( ( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - p T^{2} + p^{2} T^{4} )^{4} \) |
| 83 | \( ( 1 - 158 T^{2} + 18075 T^{4} - 158 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 89 | \( ( 1 + 18 T + p T^{2} )^{8} \) |
| 97 | \( ( 1 - 10 T + p T^{2} )^{4}( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} )^{2} \) |
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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
−3.96675960348124341205227578511, −3.90027938394681466060561491577, −3.82904029194184710066628038965, −3.64931932962850526573371533933, −3.53337978151251391886470312849, −3.45517300963357768435460630472, −2.90592525436959118530643204182, −2.88681045990556398510523355615, −2.85650846342094226684889836691, −2.71318908038735663462502077034, −2.67621424226831400482810733572, −2.59201883081624651051863911295, −2.27592102628779514045667445901, −2.16199188911955429496958102201, −2.14583958377388174977644792671, −1.99813797290488288558334319903, −1.87410509482151124497094708912, −1.64947650703116953790604918298, −1.51937619692144916135724562997, −1.34842588879978917029897579200, −1.04733544284154866620138805439, −0.957337682443293058641963987031, −0.37160976001828121923782460977, −0.31891392844759892390505726419, −0.15995126673432774885881701119,
0.15995126673432774885881701119, 0.31891392844759892390505726419, 0.37160976001828121923782460977, 0.957337682443293058641963987031, 1.04733544284154866620138805439, 1.34842588879978917029897579200, 1.51937619692144916135724562997, 1.64947650703116953790604918298, 1.87410509482151124497094708912, 1.99813797290488288558334319903, 2.14583958377388174977644792671, 2.16199188911955429496958102201, 2.27592102628779514045667445901, 2.59201883081624651051863911295, 2.67621424226831400482810733572, 2.71318908038735663462502077034, 2.85650846342094226684889836691, 2.88681045990556398510523355615, 2.90592525436959118530643204182, 3.45517300963357768435460630472, 3.53337978151251391886470312849, 3.64931932962850526573371533933, 3.82904029194184710066628038965, 3.90027938394681466060561491577, 3.96675960348124341205227578511
Plot not available for L-functions of degree greater than 10.
