| L(s) = 1 | + 3-s − 4·5-s − 9-s + 2·11-s − 5·13-s − 4·15-s − 2·17-s − 2·19-s − 2·23-s + 2·25-s + 3·29-s + 9·31-s + 2·33-s − 5·39-s − 41-s − 16·43-s + 4·45-s − 11·47-s − 2·51-s + 4·53-s − 8·55-s − 2·57-s − 4·59-s − 8·61-s + 20·65-s − 2·67-s − 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.78·5-s − 1/3·9-s + 0.603·11-s − 1.38·13-s − 1.03·15-s − 0.485·17-s − 0.458·19-s − 0.417·23-s + 2/5·25-s + 0.557·29-s + 1.61·31-s + 0.348·33-s − 0.800·39-s − 0.156·41-s − 2.43·43-s + 0.596·45-s − 1.60·47-s − 0.280·51-s + 0.549·53-s − 1.07·55-s − 0.264·57-s − 0.520·59-s − 1.02·61-s + 2.48·65-s − 0.244·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81288256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - T + 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 5 T + 28 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 2 T + 18 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 3 T + 56 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 9 T + 78 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T - 24 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 11 T + 86 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 2 T + 118 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 23 T + 270 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 17 T + 180 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2 T + 142 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 12 T + 134 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 2 T + 26 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 2 T + 42 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
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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
−7.56330079341282702145503201260, −7.40724127328967804618636481709, −6.86289216192958965420015074917, −6.63825013320547427786612851053, −6.33619493017487252300984219655, −6.04000257018911736907766435712, −5.27708529078225922263777568788, −5.02459685222775273473865310772, −4.68157053127263129910380700589, −4.40745828546534688848830949197, −4.03125056564964993871446655503, −3.57110675568849706855541784586, −3.32673412173474320840442251589, −3.00885535579403076146024760966, −2.31754051561005294562576380789, −2.21852514605938894889584562972, −1.52876411243811213366519408408, −0.847541781786928836007947699333, 0, 0,
0.847541781786928836007947699333, 1.52876411243811213366519408408, 2.21852514605938894889584562972, 2.31754051561005294562576380789, 3.00885535579403076146024760966, 3.32673412173474320840442251589, 3.57110675568849706855541784586, 4.03125056564964993871446655503, 4.40745828546534688848830949197, 4.68157053127263129910380700589, 5.02459685222775273473865310772, 5.27708529078225922263777568788, 6.04000257018911736907766435712, 6.33619493017487252300984219655, 6.63825013320547427786612851053, 6.86289216192958965420015074917, 7.40724127328967804618636481709, 7.56330079341282702145503201260
