| L(s) = 1 | − 2·7-s − 4·11-s + 4·13-s − 8·17-s − 4·23-s + 2·25-s − 4·29-s − 8·31-s − 4·37-s − 16·41-s − 16·43-s − 8·47-s + 3·49-s + 4·53-s + 16·59-s − 4·61-s + 8·67-s − 12·71-s + 12·73-s + 8·77-s − 8·79-s + 8·83-s − 8·91-s − 4·97-s + 24·101-s − 8·103-s − 4·107-s + ⋯ |
| L(s) = 1 | − 0.755·7-s − 1.20·11-s + 1.10·13-s − 1.94·17-s − 0.834·23-s + 2/5·25-s − 0.742·29-s − 1.43·31-s − 0.657·37-s − 2.49·41-s − 2.43·43-s − 1.16·47-s + 3/7·49-s + 0.549·53-s + 2.08·59-s − 0.512·61-s + 0.977·67-s − 1.42·71-s + 1.40·73-s + 0.911·77-s − 0.900·79-s + 0.878·83-s − 0.838·91-s − 0.406·97-s + 2.38·101-s − 0.788·103-s − 0.386·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 38 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4 T + 38 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 14 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 + 16 T + 134 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 59 | $D_{4}$ | \( 1 - 16 T + 134 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 102 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 12 T + 166 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 134 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 126 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 166 T^{2} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.739631925333111836237839857925, −8.656184697754643218881139649736, −8.220753089012001960091543504461, −8.000521700962134445842300853403, −7.18413625791915182051649066521, −6.95890747900309698060611027575, −6.65403836133598310184298690507, −6.31538822260685431584255654301, −5.65717690783969533954178756643, −5.44842329645105456606980467346, −4.87118591991587156534263542030, −4.60732286068367472726877166950, −3.76196897364490227476102123503, −3.52485542500822595117488446596, −3.25827446990449304549497071555, −2.28489210301317985037739261092, −2.10908433923762241202905681282, −1.42337761648183598723318652687, 0, 0,
1.42337761648183598723318652687, 2.10908433923762241202905681282, 2.28489210301317985037739261092, 3.25827446990449304549497071555, 3.52485542500822595117488446596, 3.76196897364490227476102123503, 4.60732286068367472726877166950, 4.87118591991587156534263542030, 5.44842329645105456606980467346, 5.65717690783969533954178756643, 6.31538822260685431584255654301, 6.65403836133598310184298690507, 6.95890747900309698060611027575, 7.18413625791915182051649066521, 8.000521700962134445842300853403, 8.220753089012001960091543504461, 8.656184697754643218881139649736, 8.739631925333111836237839857925
