L(s) = 1 | − 0.284·3-s + 4-s + 1.68·5-s − 0.918·9-s − 0.284·12-s − 0.478·15-s + 16-s + 1.68·20-s − 1.30·23-s + 1.83·25-s + 0.546·27-s − 1.30·31-s − 0.918·36-s − 1.91·37-s − 1.54·45-s − 1.91·47-s − 0.284·48-s + 49-s + 0.830·53-s + 0.830·59-s − 0.478·60-s + 64-s + 1.68·67-s + 0.372·69-s − 1.91·71-s − 0.521·75-s + 1.68·80-s + ⋯ |
L(s) = 1 | − 0.284·3-s + 4-s + 1.68·5-s − 0.918·9-s − 0.284·12-s − 0.478·15-s + 16-s + 1.68·20-s − 1.30·23-s + 1.83·25-s + 0.546·27-s − 1.30·31-s − 0.918·36-s − 1.91·37-s − 1.54·45-s − 1.91·47-s − 0.284·48-s + 49-s + 0.830·53-s + 0.830·59-s − 0.478·60-s + 64-s + 1.68·67-s + 0.372·69-s − 1.91·71-s − 0.521·75-s + 1.68·80-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1331 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.493124123\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.493124123\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 + 0.284T + T^{2} \) |
| 5 | \( 1 - 1.68T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + 1.30T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + 1.30T + T^{2} \) |
| 37 | \( 1 + 1.91T + T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 1.91T + T^{2} \) |
| 53 | \( 1 - 0.830T + T^{2} \) |
| 59 | \( 1 - 0.830T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.68T + T^{2} \) |
| 71 | \( 1 + 1.91T + T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - 0.830T + T^{2} \) |
| 97 | \( 1 + 1.30T + T^{2} \) |
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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
−10.04996701307171086652187515922, −9.080052118416553858891726692992, −8.283679414806592139847998311145, −7.12162294430811113903250184736, −6.38789817369931773578177745770, −5.71827764516704864163776200226, −5.23043253798328927921797341005, −3.50007564790433158306684995835, −2.40686196697843757983774320205, −1.70626672546393092902051863032,
1.70626672546393092902051863032, 2.40686196697843757983774320205, 3.50007564790433158306684995835, 5.23043253798328927921797341005, 5.71827764516704864163776200226, 6.38789817369931773578177745770, 7.12162294430811113903250184736, 8.283679414806592139847998311145, 9.080052118416553858891726692992, 10.04996701307171086652187515922