| L(s) = 1 | + 3-s − 8·7-s + 3·9-s − 6·11-s + 2·13-s + 6·17-s + 7·19-s − 8·21-s + 6·23-s + 5·25-s + 8·27-s + 4·31-s − 6·33-s + 20·37-s + 2·39-s − 9·41-s − 4·43-s + 34·49-s + 6·51-s + 6·53-s + 7·57-s − 9·59-s − 4·61-s − 24·63-s − 7·67-s + 6·69-s + 6·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 3.02·7-s + 9-s − 1.80·11-s + 0.554·13-s + 1.45·17-s + 1.60·19-s − 1.74·21-s + 1.25·23-s + 25-s + 1.53·27-s + 0.718·31-s − 1.04·33-s + 3.28·37-s + 0.320·39-s − 1.40·41-s − 0.609·43-s + 34/7·49-s + 0.840·51-s + 0.824·53-s + 0.927·57-s − 1.17·59-s − 0.512·61-s − 3.02·63-s − 0.855·67-s + 0.722·69-s + 0.712·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.993492983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.993492983\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 9 T + 40 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 4 T - 45 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - T - 72 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 17 T + 192 T^{2} + 17 p T^{3} + p^{2} T^{4} \) |
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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
−9.892050146137477846198667778016, −9.648076560967476764093598862214, −9.248943792876180282106526506103, −8.888678574573461179085764885642, −8.204052810644402587360936798804, −7.900225712865470621351691421551, −7.44424171897927928764187474818, −7.05006633339860145427753795746, −6.63394717743279723533903728594, −6.35575204356113544667431827066, −5.65396617700276012431969186488, −5.48198430761843276797236102550, −4.76468475562860726325209260769, −4.29876058044247304345007542289, −3.35407486990990152512571686815, −3.29096627504812227289805055781, −2.78225318144012742940223962659, −2.72847360440711388628494488929, −1.21483248383309752623603914887, −0.66564732616554931686820538104,
0.66564732616554931686820538104, 1.21483248383309752623603914887, 2.72847360440711388628494488929, 2.78225318144012742940223962659, 3.29096627504812227289805055781, 3.35407486990990152512571686815, 4.29876058044247304345007542289, 4.76468475562860726325209260769, 5.48198430761843276797236102550, 5.65396617700276012431969186488, 6.35575204356113544667431827066, 6.63394717743279723533903728594, 7.05006633339860145427753795746, 7.44424171897927928764187474818, 7.900225712865470621351691421551, 8.204052810644402587360936798804, 8.888678574573461179085764885642, 9.248943792876180282106526506103, 9.648076560967476764093598862214, 9.892050146137477846198667778016
