| L(s) = 1 | + 2·7-s − 4·17-s + 12·23-s + 6·25-s + 8·31-s − 20·41-s − 16·47-s + 3·49-s − 4·71-s − 12·73-s + 16·79-s + 4·89-s + 4·97-s + 16·103-s + 32·113-s − 8·119-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 24·161-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | + 0.755·7-s − 0.970·17-s + 2.50·23-s + 6/5·25-s + 1.43·31-s − 3.12·41-s − 2.33·47-s + 3/7·49-s − 0.474·71-s − 1.40·73-s + 1.80·79-s + 0.423·89-s + 0.406·97-s + 1.57·103-s + 3.01·113-s − 0.733·119-s + 6/11·121-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 + 1.89·161-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16257024 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.103793714\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.103793714\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 54 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 82 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 102 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.639875481829791200013892669803, −8.442371547175336999943056837227, −7.86314905279729202615487355783, −7.59638880887042968116801712542, −7.03056461178261996756527945943, −6.81131407249797504257193690175, −6.48185555575922770787078390307, −6.28512407886375264188492068824, −5.35243811599454770504969388688, −5.31610857572994125748825871572, −4.71141296883123168703497550450, −4.69085563844256597879338659068, −4.28608044658885924417009042248, −3.36163048127631847066311575155, −3.10736478717372928895268746767, −3.00477697946374691280068410401, −1.94151499877687298318719051594, −1.90487483604807027648049637196, −1.09453873448597372328137395150, −0.56494263130347374654485902062,
0.56494263130347374654485902062, 1.09453873448597372328137395150, 1.90487483604807027648049637196, 1.94151499877687298318719051594, 3.00477697946374691280068410401, 3.10736478717372928895268746767, 3.36163048127631847066311575155, 4.28608044658885924417009042248, 4.69085563844256597879338659068, 4.71141296883123168703497550450, 5.31610857572994125748825871572, 5.35243811599454770504969388688, 6.28512407886375264188492068824, 6.48185555575922770787078390307, 6.81131407249797504257193690175, 7.03056461178261996756527945943, 7.59638880887042968116801712542, 7.86314905279729202615487355783, 8.442371547175336999943056837227, 8.639875481829791200013892669803
