| L(s) = 1 | + 2·9-s + 20·25-s + 36·29-s − 28·49-s + 19·81-s − 44·109-s + 26·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 38·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 40·225-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 4·25-s + 6.68·29-s − 4·49-s + 19/9·81-s − 4.21·109-s + 2.36·121-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 − 2.92·169-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 + 0.0669·223-s + 8/3·225-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 5^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{8} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.759533132\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.759533132\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( ( 1 - p T^{2} )^{4} \) |
| 7 | \( ( 1 + p T^{2} )^{4} \) |
| good | 3 | \( ( 1 - T^{2} - 8 T^{4} - p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 11 | \( ( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2}( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | \( ( 1 + 19 T^{2} + 192 T^{4} + 19 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 17 | \( ( 1 - 29 T^{2} + 552 T^{4} - 29 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 19 | \( ( 1 + p T^{2} )^{8} \) |
| 23 | \( ( 1 + p T^{2} )^{8} \) |
| 29 | \( ( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} )^{4} \) |
| 31 | \( ( 1 + p T^{2} )^{8} \) |
| 37 | \( ( 1 - p T^{2} )^{8} \) |
| 41 | \( ( 1 - p T^{2} )^{8} \) |
| 43 | \( ( 1 + p T^{2} )^{8} \) |
| 47 | \( ( 1 + 31 T^{2} - 1248 T^{4} + 31 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
| 53 | \( ( 1 - p T^{2} )^{8} \) |
| 59 | \( ( 1 + p T^{2} )^{8} \) |
| 61 | \( ( 1 - p T^{2} )^{8} \) |
| 67 | \( ( 1 + p T^{2} )^{8} \) |
| 71 | \( ( 1 - 12 T + p T^{2} )^{4}( 1 + 12 T + p T^{2} )^{4} \) |
| 73 | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{4} \) |
| 79 | \( ( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} )^{2}( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{4} \) |
| 89 | \( ( 1 - p T^{2} )^{8} \) |
| 97 | \( ( 1 - 149 T^{2} + 12792 T^{4} - 149 p^{2} T^{6} + p^{4} T^{8} )^{2} \) |
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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
−4.76654256241646054048555899341, −4.49018042332909453534590621514, −4.45091197847790903113839289759, −4.42283328646971430529917934844, −4.33263776236038357050328031121, −4.08095855250469766737534954600, −3.83351106868102674910036791202, −3.82473895118450275634240307208, −3.35919897074833712652306823351, −3.26607130526066882779351456483, −3.25917129190739408353162167940, −3.13638647498703291988239485388, −2.88727573701469633190227850177, −2.78001999655357903604867306239, −2.77787352405206109954398425949, −2.48170414735162177759869757300, −2.44320713941922778342131863683, −2.00604304111960280139161345393, −1.80688492210562905965223826298, −1.71866435994139693634324893874, −1.26978788452515273596639357370, −1.12129596075835381195169056338, −1.00578099066002059224857071411, −0.865849303100273279336043796972, −0.42311400585236984447762164787,
0.42311400585236984447762164787, 0.865849303100273279336043796972, 1.00578099066002059224857071411, 1.12129596075835381195169056338, 1.26978788452515273596639357370, 1.71866435994139693634324893874, 1.80688492210562905965223826298, 2.00604304111960280139161345393, 2.44320713941922778342131863683, 2.48170414735162177759869757300, 2.77787352405206109954398425949, 2.78001999655357903604867306239, 2.88727573701469633190227850177, 3.13638647498703291988239485388, 3.25917129190739408353162167940, 3.26607130526066882779351456483, 3.35919897074833712652306823351, 3.82473895118450275634240307208, 3.83351106868102674910036791202, 4.08095855250469766737534954600, 4.33263776236038357050328031121, 4.42283328646971430529917934844, 4.45091197847790903113839289759, 4.49018042332909453534590621514, 4.76654256241646054048555899341
Plot not available for L-functions of degree greater than 10.
