| L(s) = 1 | + 4·7-s − 3·9-s − 10·25-s + 20·37-s − 16·43-s + 9·49-s − 12·63-s − 32·67-s − 8·79-s + 9·81-s − 4·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 22·169-s + 173-s − 40·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | + 1.51·7-s − 9-s − 2·25-s + 3.28·37-s − 2.43·43-s + 9/7·49-s − 1.51·63-s − 3.90·67-s − 0.900·79-s + 81-s − 0.383·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 1.69·169-s + 0.0760·173-s − 3.02·175-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9653431326\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9653431326\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + p T^{2} \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−14.80511695638160694849422515926, −14.01514784981660746800938046652, −13.45709188274793648937010381999, −13.24271376041736625582065078385, −12.11078666738757929154452669387, −11.81863185793182487827709098494, −11.29182449378243979210408310932, −11.03872821307273511904902517339, −10.13000869513673846219308250556, −9.626848095956417877996067048542, −8.860431203632680285379101442976, −8.231499877023010211585730486077, −7.889190764798384171636941586833, −7.27688550516460778984917889313, −6.07005500345546038890961276394, −5.80122190357005510453887893234, −4.81237983127692909102423173270, −4.24043619370547132373815289388, −3.01248503846707493834137147349, −1.84955169590426068716106906195,
1.84955169590426068716106906195, 3.01248503846707493834137147349, 4.24043619370547132373815289388, 4.81237983127692909102423173270, 5.80122190357005510453887893234, 6.07005500345546038890961276394, 7.27688550516460778984917889313, 7.889190764798384171636941586833, 8.231499877023010211585730486077, 8.860431203632680285379101442976, 9.626848095956417877996067048542, 10.13000869513673846219308250556, 11.03872821307273511904902517339, 11.29182449378243979210408310932, 11.81863185793182487827709098494, 12.11078666738757929154452669387, 13.24271376041736625582065078385, 13.45709188274793648937010381999, 14.01514784981660746800938046652, 14.80511695638160694849422515926
