L(s) = 1 | − 2.23i·2-s + 3-s − 3.00·4-s + 3·5-s − 2.23i·6-s + 2.23i·8-s − 2·9-s − 6.70i·10-s + 3·11-s − 3.00·12-s + 3·15-s − 0.999·16-s − 3·17-s + 4.47i·18-s − 19-s − 9.00·20-s + ⋯ |
L(s) = 1 | − 1.58i·2-s + 0.577·3-s − 1.50·4-s + 1.34·5-s − 0.912i·6-s + 0.790i·8-s − 0.666·9-s − 2.12i·10-s + 0.904·11-s − 0.866·12-s + 0.774·15-s − 0.249·16-s − 0.727·17-s + 1.05i·18-s − 0.229·19-s − 2.01·20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.462 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.827711 - 1.36545i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.827711 - 1.36545i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 229 | \( 1 + (-7 + 13.4i)T \) |
good | 2 | \( 1 + 2.23iT - 2T^{2} \) |
| 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 - 3T + 5T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 + 3T + 17T^{2} \) |
| 19 | \( 1 + T + 19T^{2} \) |
| 23 | \( 1 - 4.47iT - 23T^{2} \) |
| 29 | \( 1 - 4.47iT - 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 - 4.47iT - 41T^{2} \) |
| 43 | \( 1 + T + 43T^{2} \) |
| 47 | \( 1 + 8.94iT - 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 5T + 61T^{2} \) |
| 67 | \( 1 + 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 13.4iT - 73T^{2} \) |
| 79 | \( 1 - 13.4iT - 79T^{2} \) |
| 83 | \( 1 + 9T + 83T^{2} \) |
| 89 | \( 1 - 17.8iT - 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−11.75485079284358130848965550474, −10.99769158155824883235202256723, −9.933228883600645345117312716392, −9.288995754727251390007447684528, −8.587851934459013834761684572790, −6.71778066430661336888582716183, −5.42927634164617286025699613855, −3.88261708380527167211809567495, −2.66907075111205383917085810778, −1.65730355152551961546014863016,
2.37938663941908880657930341002, 4.38492488245376512643072179509, 5.78640405170751959347105161981, 6.29205951881109080234144756635, 7.41739122861685143280509277723, 8.768961470526460026351817099329, 8.998106452521941953152706182178, 10.20704899738732125371168016271, 11.61430493196621781706902305185, 13.14580004509351118819641553580