| L(s) = 1 | + 2·3-s − 2·5-s − 2·7-s + 3·9-s − 4·13-s − 4·15-s + 4·17-s − 4·21-s − 8·23-s + 3·25-s + 4·27-s − 4·29-s + 4·35-s − 12·37-s − 8·39-s − 4·41-s − 8·43-s − 6·45-s + 3·49-s + 8·51-s + 4·53-s − 4·61-s − 6·63-s + 8·65-s − 8·67-s − 16·69-s − 16·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.755·7-s + 9-s − 1.10·13-s − 1.03·15-s + 0.970·17-s − 0.872·21-s − 1.66·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s + 0.676·35-s − 1.97·37-s − 1.28·39-s − 0.624·41-s − 1.21·43-s − 0.894·45-s + 3/7·49-s + 1.12·51-s + 0.549·53-s − 0.512·61-s − 0.755·63-s + 0.992·65-s − 0.977·67-s − 1.92·69-s − 1.89·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11289600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 11 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_4$ | \( 1 - 4 T + 6 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 54 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + 4 T - 2 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 198 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 8 T + 46 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 24 T + 278 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 20 T + 286 T^{2} + 20 p T^{3} + p^{2} T^{4} \) |
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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
−8.341815973234334686735013134223, −8.222497847810269173786429259534, −7.56207639966413176980451987651, −7.43087352392658665501795336145, −7.11602385877386513105320175328, −6.80655579085140909802627829783, −6.21908520510981204691000830052, −5.78023573032949690179977793912, −5.37674728200089548264560618525, −4.93472125502222888928265029363, −4.37434062880841368629807363854, −4.07449901420075033999777888130, −3.54408738565499862211590341845, −3.41954278715241122123657569141, −2.66711578800849608008231723759, −2.66424027348982639427096626958, −1.57273148498498219763634975285, −1.56116966496114657239029463648, 0, 0,
1.56116966496114657239029463648, 1.57273148498498219763634975285, 2.66424027348982639427096626958, 2.66711578800849608008231723759, 3.41954278715241122123657569141, 3.54408738565499862211590341845, 4.07449901420075033999777888130, 4.37434062880841368629807363854, 4.93472125502222888928265029363, 5.37674728200089548264560618525, 5.78023573032949690179977793912, 6.21908520510981204691000830052, 6.80655579085140909802627829783, 7.11602385877386513105320175328, 7.43087352392658665501795336145, 7.56207639966413176980451987651, 8.222497847810269173786429259534, 8.341815973234334686735013134223
