| 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 + 2·25-s + 3·29-s − 9·31-s + 2·33-s − 5·39-s + 41-s + 16·43-s + 4·45-s + 11·47-s − 14·49-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 + 23·71-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 + 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 − 2·49-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 + 2.72·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 17909824 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 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
−8.326626633220670777821405331064, −7.65315077158478791595437081907, −7.59152262556407002107682340587, −7.29475820678679649379623887854, −6.83928833019883084405933779020, −6.16797119356373253979552606937, −6.00273049897016028564963164862, −5.86677732344629980332099528824, −5.12537462782071575462476159128, −4.87300842899089915883733865559, −4.34953693475339545572915495611, −3.96225879966067806906472088880, −3.62832499751796219873646833265, −3.50038521539427304027752644448, −2.57904717641953780237917430349, −2.42681232437533023271499904385, −1.53380455915213139797559434035, −0.965204595135563988441797034842, 0, 0,
0.965204595135563988441797034842, 1.53380455915213139797559434035, 2.42681232437533023271499904385, 2.57904717641953780237917430349, 3.50038521539427304027752644448, 3.62832499751796219873646833265, 3.96225879966067806906472088880, 4.34953693475339545572915495611, 4.87300842899089915883733865559, 5.12537462782071575462476159128, 5.86677732344629980332099528824, 6.00273049897016028564963164862, 6.16797119356373253979552606937, 6.83928833019883084405933779020, 7.29475820678679649379623887854, 7.59152262556407002107682340587, 7.65315077158478791595437081907, 8.326626633220670777821405331064
