L(s) = 1 | + 2-s + 1.41i·5-s − 7-s − 8-s − 9-s + 1.41i·10-s − 1.41i·11-s − 13-s − 14-s − 16-s + 1.41i·17-s − 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯ |
L(s) = 1 | + 2-s + 1.41i·5-s − 7-s − 8-s − 9-s + 1.41i·10-s − 1.41i·11-s − 13-s − 14-s − 16-s + 1.41i·17-s − 18-s − 1.41i·19-s − 1.41i·22-s − 1.00·25-s − 26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3284938577\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3284938577\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 - T \) |
| 59 | \( 1 + T \) |
good | 2 | \( 1 - T + T^{2} \) |
| 3 | \( 1 + T^{2} \) |
| 5 | \( 1 - 1.41iT - T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + 1.41iT - T^{2} \) |
| 13 | \( 1 + T + T^{2} \) |
| 17 | \( 1 - 1.41iT - T^{2} \) |
| 19 | \( 1 + 1.41iT - T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.41iT - T^{2} \) |
| 31 | \( 1 + T + T^{2} \) |
| 41 | \( 1 + T + T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - 1.41iT - T^{2} \) |
| 53 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T + T^{2} \) |
| 67 | \( 1 + 1.41iT - T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 - 1.41iT - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T + T^{2} \) |
| 97 | \( 1 - T + T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.532465787475984365750170095985, −8.973596509034156032405431219345, −8.067692065930665380064693294736, −6.92197854746022076756463447281, −6.32617190487885466280737279220, −5.82966402251187479712063556801, −4.91271949515141280751528950564, −3.61310997320949362430893589971, −3.14290753438165019520082492940, −2.61822302852796594794184042386,
0.14930695123441591838264459465, 2.15574307593812869283521030604, 3.16925160918246236881168404631, 4.17968671018951219770062979042, 4.87764311277989735638761271227, 5.43238234926388620078296698820, 6.19714624328991293007348906898, 7.23854201772672234103692300408, 8.122234452757815685402598422198, 9.037722429598036170613219066183