| L(s) = 1 | + 2-s + 2·4-s − 5-s + 7-s + 5·8-s − 10-s + 2·11-s + 5·13-s + 14-s + 5·16-s + 6·17-s − 4·19-s − 2·20-s + 2·22-s − 6·23-s + 5·25-s + 5·26-s + 2·28-s + 5·29-s + 6·31-s + 10·32-s + 6·34-s − 35-s − 6·37-s − 4·38-s − 5·40-s + 10·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 4-s − 0.447·5-s + 0.377·7-s + 1.76·8-s − 0.316·10-s + 0.603·11-s + 1.38·13-s + 0.267·14-s + 5/4·16-s + 1.45·17-s − 0.917·19-s − 0.447·20-s + 0.426·22-s − 1.25·23-s + 25-s + 0.980·26-s + 0.377·28-s + 0.928·29-s + 1.07·31-s + 1.76·32-s + 1.02·34-s − 0.169·35-s − 0.986·37-s − 0.648·38-s − 0.790·40-s + 1.56·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.150785890\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.150785890\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + T - 4 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T - 7 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 5 T - 4 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 10 T + 59 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 7 T - 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 2 T - 63 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 18 T + 241 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 18 T + 227 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
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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
−10.82768414564893079969128086914, −10.75485500072417480909024911787, −10.15600469758599658379462209532, −10.03996502365996857752414646311, −9.044318894565695844381424064002, −8.674836319106929716571124931836, −8.199022846786281880979328162591, −7.82915392469211992439557756581, −7.29995247329131200860920694018, −7.01249251634111514600746523294, −6.14323805117760070964408016096, −6.10074926404431677972302125838, −5.56458345590890897927138428658, −4.54492900747510792087865449172, −4.39511277959102820086688601117, −4.02996872807467022844668338167, −3.10164539934725013716210561112, −2.77519122003841945282314792036, −1.49626899386113813250961403039, −1.35838487573151142096633233779,
1.35838487573151142096633233779, 1.49626899386113813250961403039, 2.77519122003841945282314792036, 3.10164539934725013716210561112, 4.02996872807467022844668338167, 4.39511277959102820086688601117, 4.54492900747510792087865449172, 5.56458345590890897927138428658, 6.10074926404431677972302125838, 6.14323805117760070964408016096, 7.01249251634111514600746523294, 7.29995247329131200860920694018, 7.82915392469211992439557756581, 8.199022846786281880979328162591, 8.674836319106929716571124931836, 9.044318894565695844381424064002, 10.03996502365996857752414646311, 10.15600469758599658379462209532, 10.75485500072417480909024911787, 10.82768414564893079969128086914
