| L(s) = 1 | + 2·7-s − 25-s + 2·37-s − 2·43-s + 3·49-s − 4·67-s + 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | + 2·7-s − 25-s + 2·37-s − 2·43-s + 3·49-s − 4·67-s + 2·79-s + 2·109-s − 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + 173-s − 2·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9144576 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.789986775\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.789986775\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_1$ | \( ( 1 + T )^{4} \) |
| 71 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 73 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
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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
−8.971434214457733538101075978741, −8.749545310839394340852514260220, −8.225444421599884955409110281250, −7.956956045757634347067280164300, −7.56732748765223462361107899286, −7.54683341707472553299442476092, −6.88813098483145403202263506948, −6.33423965761509016226875554844, −6.12544589897098608098052640703, −5.43591481680323737437400003679, −5.40786659483238764739364568965, −4.71328834798529393613601967817, −4.47784437848657049244926002449, −4.20440043296252529937990751821, −3.57530321179120267223869669870, −3.03162559877840238268089610469, −2.51226482984222589948996111273, −1.78692070865236446985622004231, −1.71811150765318062888341207723, −0.874020922461783655717218912061,
0.874020922461783655717218912061, 1.71811150765318062888341207723, 1.78692070865236446985622004231, 2.51226482984222589948996111273, 3.03162559877840238268089610469, 3.57530321179120267223869669870, 4.20440043296252529937990751821, 4.47784437848657049244926002449, 4.71328834798529393613601967817, 5.40786659483238764739364568965, 5.43591481680323737437400003679, 6.12544589897098608098052640703, 6.33423965761509016226875554844, 6.88813098483145403202263506948, 7.54683341707472553299442476092, 7.56732748765223462361107899286, 7.956956045757634347067280164300, 8.225444421599884955409110281250, 8.749545310839394340852514260220, 8.971434214457733538101075978741
