| L(s) = 1 | − 8·7-s + 4·17-s − 8·23-s + 6·25-s − 12·41-s − 16·47-s + 34·49-s − 24·71-s − 28·73-s + 16·79-s − 4·89-s − 4·97-s − 8·103-s − 4·113-s − 32·119-s + 18·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 64·161-s + 163-s + 167-s + 22·169-s + ⋯ |
| L(s) = 1 | − 3.02·7-s + 0.970·17-s − 1.66·23-s + 6/5·25-s − 1.87·41-s − 2.33·47-s + 34/7·49-s − 2.84·71-s − 3.27·73-s + 1.80·79-s − 0.423·89-s − 0.406·97-s − 0.788·103-s − 0.376·113-s − 2.93·119-s + 1.63·121-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 + 5.04·161-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5308416 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 18 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 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
−8.901283822911148933219274961961, −8.537883598187149394139660939031, −7.927597082960768217010357813251, −7.78807401181537754550940713269, −6.90658693472445454577262700351, −6.88483364047604802017468384205, −6.63217600020591710133260327071, −6.05413092799802394714867394531, −5.75245531885267310765400026046, −5.54545642580210773785881917936, −4.65041292432824869473491905227, −4.43840339046971541107811169446, −3.54504386175201089083978945511, −3.51512988593556574794619540593, −3.02134894909850940868110394782, −2.76949698107409344784867387252, −1.89489278127510207413757070938, −1.21146561810893222769175172833, 0, 0,
1.21146561810893222769175172833, 1.89489278127510207413757070938, 2.76949698107409344784867387252, 3.02134894909850940868110394782, 3.51512988593556574794619540593, 3.54504386175201089083978945511, 4.43840339046971541107811169446, 4.65041292432824869473491905227, 5.54545642580210773785881917936, 5.75245531885267310765400026046, 6.05413092799802394714867394531, 6.63217600020591710133260327071, 6.88483364047604802017468384205, 6.90658693472445454577262700351, 7.78807401181537754550940713269, 7.927597082960768217010357813251, 8.537883598187149394139660939031, 8.901283822911148933219274961961
