| L(s) = 1 | + 4·2-s + 8·4-s − 4·5-s − 4·7-s + 8·8-s − 16·10-s + 16·11-s − 16·14-s − 4·16-s − 32·20-s + 64·22-s + 8·25-s − 32·28-s − 32·32-s + 16·35-s − 32·40-s + 128·44-s + 8·49-s + 32·50-s − 64·55-s − 32·56-s − 64·64-s + 64·70-s + 28·73-s − 64·77-s + 16·80-s − 9·81-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 4·4-s − 1.78·5-s − 1.51·7-s + 2.82·8-s − 5.05·10-s + 4.82·11-s − 4.27·14-s − 16-s − 7.15·20-s + 13.6·22-s + 8/5·25-s − 6.04·28-s − 5.65·32-s + 2.70·35-s − 5.05·40-s + 19.2·44-s + 8/7·49-s + 4.52·50-s − 8.62·55-s − 4.27·56-s − 8·64-s + 7.64·70-s + 3.27·73-s − 7.29·77-s + 1.78·80-s − 81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.318297227\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.318297227\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( ( 1 - p T + p T^{2} )^{2} \) |
| 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 7 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 10 T^{2} + p^{2} T^{4} ) \) |
| 11 | $C_2^2$ | \( ( 1 - 8 T + 32 T^{2} - 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 4 T + 8 T^{2} - 4 p T^{3} + p^{2} T^{4} )( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} ) \) |
| 31 | $C_2^2$ | \( ( 1 + 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 20 T + 200 T^{2} - 20 p T^{3} + p^{2} T^{4} )( 1 + 20 T + 200 T^{2} + 20 p T^{3} + p^{2} T^{4} ) \) |
| 59 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 16 T + 128 T^{2} - 16 p T^{3} + p^{2} T^{4} )( 1 + 16 T + 128 T^{2} + 16 p T^{3} + p^{2} T^{4} ) \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 67 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - 14 T + p T^{2} )^{2}( 1 + 50 T^{2} + p^{2} T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2}( 1 + 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 + 8 T + 32 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 + 2 T + p T^{2} )^{2}( 1 - 190 T^{2} + p^{2} T^{4} ) \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.935841918997355946545608675042, −9.416200152755732205001827323502, −9.364097183916475805325918138898, −9.187057553234918187388590182671, −8.947596864384573830521560302011, −8.521603896613374908086709270396, −8.397913646367035705187934380753, −7.67495529477065139403961646919, −7.34089307189216766017915737471, −6.97958241767360526678872569909, −6.88596866754836332506236089286, −6.43167306460902167475225756258, −6.28034123102157793179296107663, −6.22322758073195348110990522134, −5.87515932874913272184166221762, −5.11850981422173399394926350098, −4.85714829861384773060946604711, −4.54417777661049656713471694880, −3.90086673055437579412211919535, −3.78877405540473134226226879488, −3.72901146605040982718353943067, −3.71876007951099834816013755585, −2.99776426019454343936143168879, −2.44366999295522310192634292701, −1.33236766640431911624838688485,
1.33236766640431911624838688485, 2.44366999295522310192634292701, 2.99776426019454343936143168879, 3.71876007951099834816013755585, 3.72901146605040982718353943067, 3.78877405540473134226226879488, 3.90086673055437579412211919535, 4.54417777661049656713471694880, 4.85714829861384773060946604711, 5.11850981422173399394926350098, 5.87515932874913272184166221762, 6.22322758073195348110990522134, 6.28034123102157793179296107663, 6.43167306460902167475225756258, 6.88596866754836332506236089286, 6.97958241767360526678872569909, 7.34089307189216766017915737471, 7.67495529477065139403961646919, 8.397913646367035705187934380753, 8.521603896613374908086709270396, 8.947596864384573830521560302011, 9.187057553234918187388590182671, 9.364097183916475805325918138898, 9.416200152755732205001827323502, 9.935841918997355946545608675042
