L(s) = 1 | + (−1 − i)2-s + 2i·4-s + (0.707 + 2.12i)5-s + (2 − 2i)8-s − 3i·9-s + (1.41 − 2.82i)10-s + (2.82 − 2.82i)13-s − 4·16-s + (1.41 + 1.41i)17-s + (−3 + 3i)18-s + (−4.24 + 1.41i)20-s + (−3.99 + 3i)25-s − 5.65·26-s − 10i·29-s + (4 + 4i)32-s + ⋯ |
L(s) = 1 | + (−0.707 − 0.707i)2-s + i·4-s + (0.316 + 0.948i)5-s + (0.707 − 0.707i)8-s − i·9-s + (0.447 − 0.894i)10-s + (0.784 − 0.784i)13-s − 16-s + (0.342 + 0.342i)17-s + (−0.707 + 0.707i)18-s + (−0.948 + 0.316i)20-s + (−0.799 + 0.600i)25-s − 1.10·26-s − 1.85i·29-s + (0.707 + 0.707i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.640i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13394 - 0.411015i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13394 - 0.411015i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + i)T \) |
| 5 | \( 1 + (-0.707 - 2.12i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + 3iT^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 + (-2.82 + 2.82i)T - 13iT^{2} \) |
| 17 | \( 1 + (-1.41 - 1.41i)T + 17iT^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + 10iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 + (-7 - 7i)T + 37iT^{2} \) |
| 41 | \( 1 - 12.7T + 41T^{2} \) |
| 43 | \( 1 + 43iT^{2} \) |
| 47 | \( 1 - 47iT^{2} \) |
| 53 | \( 1 + (5 - 5i)T - 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 15.5T + 61T^{2} \) |
| 67 | \( 1 - 67iT^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-11.3 + 11.3i)T - 73iT^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 83iT^{2} \) |
| 89 | \( 1 + 4.24iT - 89T^{2} \) |
| 97 | \( 1 + (-12.7 - 12.7i)T + 97iT^{2} \) |
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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
−9.842760372448919755619552293834, −9.407599741721911504102023546617, −8.249387227602814150297796068867, −7.63768115461945467205999673119, −6.51042731693190920028875215951, −5.91444399698183451301738560836, −4.15440872230415923795123822006, −3.33305537992042186804111962294, −2.43706275343250936023288442770, −0.917732780764235464981810779962,
1.09983619476584103195423656791, 2.21525845414716319568717366953, 4.17017338885576154380246057238, 5.12383023234967833578587191681, 5.76061917579928032392477690676, 6.81034006875914318767744168756, 7.72722586867277790912998920717, 8.458274548158230114709519922458, 9.157978086876935646706110727640, 9.789873628413261839453366404746