| L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 2·8-s − 2·9-s − 2·13-s − 4·14-s − 4·16-s − 4·18-s − 4·26-s − 2·28-s − 2·31-s − 2·32-s − 2·36-s + 2·37-s − 2·41-s + 49-s − 2·52-s − 2·53-s + 4·56-s − 2·59-s + 2·61-s − 4·62-s + 4·63-s + 3·64-s − 2·71-s + 4·72-s + ⋯ |
| L(s) = 1 | + 2·2-s + 4-s − 2·7-s − 2·8-s − 2·9-s − 2·13-s − 4·14-s − 4·16-s − 4·18-s − 4·26-s − 2·28-s − 2·31-s − 2·32-s − 2·36-s + 2·37-s − 2·41-s + 49-s − 2·52-s − 2·53-s + 4·56-s − 2·59-s + 2·61-s − 4·62-s + 4·63-s + 3·64-s − 2·71-s + 4·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4765489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4765489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1079082145\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1079082145\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 37 | $C_1$ | \( ( 1 - T )^{2} \) |
| 59 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + T^{4} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + T^{4} \) |
| 13 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + T^{4} \) |
| 19 | $C_2^2$ | \( 1 + T^{4} \) |
| 23 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + T^{4} \) |
| 53 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−9.532465787475984365750170095985, −9.037722429598036170613219066183, −8.973596509034156032405431219345, −8.122234452757815685402598422198, −8.067692065930665380064693294736, −7.23854201772672234103692300408, −6.92197854746022076756463447281, −6.32617190487885466280737279220, −6.19714624328991293007348906898, −5.82966402251187479712063556801, −5.43238234926388620078296698820, −4.91271949515141280751528950564, −4.87764311277989735638761271227, −4.17968671018951219770062979042, −3.61310997320949362430893589971, −3.16925160918246236881168404631, −3.14290753438165019520082492940, −2.61822302852796594794184042386, −2.15574307593812869283521030604, −0.14930695123441591838264459465,
0.14930695123441591838264459465, 2.15574307593812869283521030604, 2.61822302852796594794184042386, 3.14290753438165019520082492940, 3.16925160918246236881168404631, 3.61310997320949362430893589971, 4.17968671018951219770062979042, 4.87764311277989735638761271227, 4.91271949515141280751528950564, 5.43238234926388620078296698820, 5.82966402251187479712063556801, 6.19714624328991293007348906898, 6.32617190487885466280737279220, 6.92197854746022076756463447281, 7.23854201772672234103692300408, 8.067692065930665380064693294736, 8.122234452757815685402598422198, 8.973596509034156032405431219345, 9.037722429598036170613219066183, 9.532465787475984365750170095985
