| L(s) = 1 | − 2-s − 2·4-s − 2·5-s − 4·7-s + 3·8-s − 9-s + 2·10-s − 4·11-s + 4·14-s + 16-s + 2·17-s + 18-s − 4·19-s + 4·20-s + 4·22-s + 12·23-s + 3·25-s + 8·28-s − 6·29-s − 2·32-s − 2·34-s + 8·35-s + 2·36-s − 6·37-s + 4·38-s − 6·40-s − 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 4-s − 0.894·5-s − 1.51·7-s + 1.06·8-s − 1/3·9-s + 0.632·10-s − 1.20·11-s + 1.06·14-s + 1/4·16-s + 0.485·17-s + 0.235·18-s − 0.917·19-s + 0.894·20-s + 0.852·22-s + 2.50·23-s + 3/5·25-s + 1.51·28-s − 1.11·29-s − 0.353·32-s − 0.342·34-s + 1.35·35-s + 1/3·36-s − 0.986·37-s + 0.648·38-s − 0.948·40-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 4 T + 13 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 2 T + 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 37 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 6 T + 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 6 T + 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 97 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 109 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 8 T + 105 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 153 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 + 2 T + 175 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
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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
−9.923164590698270054817650236949, −9.543327251904744593915378526396, −8.976502786094128164749537110187, −8.849061319706564619488413156442, −8.438947367349405128549731943184, −7.972570524500856780221601387272, −7.46942427363821226489207152276, −7.18310541081407035948449286458, −6.41775426363009024126125164811, −6.38227455954925179501878282735, −5.23362312287390411068686025004, −5.15525107464586761210442922481, −4.76575291969552224028215418120, −3.83728644848747355831933995873, −3.52735459674842479947563113735, −3.08380397726926203489760297355, −2.43367768257096728715119096413, −1.20424520096660075309951790827, 0, 0,
1.20424520096660075309951790827, 2.43367768257096728715119096413, 3.08380397726926203489760297355, 3.52735459674842479947563113735, 3.83728644848747355831933995873, 4.76575291969552224028215418120, 5.15525107464586761210442922481, 5.23362312287390411068686025004, 6.38227455954925179501878282735, 6.41775426363009024126125164811, 7.18310541081407035948449286458, 7.46942427363821226489207152276, 7.972570524500856780221601387272, 8.438947367349405128549731943184, 8.849061319706564619488413156442, 8.976502786094128164749537110187, 9.543327251904744593915378526396, 9.923164590698270054817650236949
