L(s) = 1 | − 0.876·2-s + 1.73·3-s − 3.23·4-s − 8.50i·5-s − 1.51·6-s − 2.64i·7-s + 6.34·8-s + 2.99·9-s + 7.46i·10-s + 12.5i·11-s − 5.59·12-s + 23.0·13-s + 2.32i·14-s − 14.7i·15-s + 7.36·16-s − 13.0i·17-s + ⋯ |
L(s) = 1 | − 0.438·2-s + 0.577·3-s − 0.807·4-s − 1.70i·5-s − 0.253·6-s − 0.377i·7-s + 0.792·8-s + 0.333·9-s + 0.746i·10-s + 1.13i·11-s − 0.466·12-s + 1.77·13-s + 0.165i·14-s − 0.982i·15-s + 0.460·16-s − 0.769i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.557 + 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.218092274\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.218092274\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - 1.73T \) |
| 7 | \( 1 + 2.64iT \) |
| 23 | \( 1 + (19.0 + 12.8i)T \) |
good | 2 | \( 1 + 0.876T + 4T^{2} \) |
| 5 | \( 1 + 8.50iT - 25T^{2} \) |
| 11 | \( 1 - 12.5iT - 121T^{2} \) |
| 13 | \( 1 - 23.0T + 169T^{2} \) |
| 17 | \( 1 + 13.0iT - 289T^{2} \) |
| 19 | \( 1 + 23.2iT - 361T^{2} \) |
| 29 | \( 1 + 13.6T + 841T^{2} \) |
| 31 | \( 1 + 23.6T + 961T^{2} \) |
| 37 | \( 1 + 38.3iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 27.0T + 1.68e3T^{2} \) |
| 43 | \( 1 - 5.04iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 21.8T + 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 50.7T + 3.48e3T^{2} \) |
| 61 | \( 1 - 68.5iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 34.3iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 103.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 128.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 130. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 39.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 5.35iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 142. iT - 9.40e3T^{2} \) |
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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
−10.07883574698764321376254886966, −9.311859440488175418009579546706, −8.765853201093634260933558054820, −8.136202346508740717529439901851, −7.10014880722105428675698956548, −5.45893037893093424620875994777, −4.53871433255096603446141287551, −3.87958344380870280069736543202, −1.72180785687874585793092410375, −0.58561838838583273683442061400,
1.67796502743113303214198683040, 3.44425323463851228529985990185, 3.70801593721315253054570229022, 5.75607076425259762067659779324, 6.40714806944518771369747570741, 7.82985066424318611935467235374, 8.304864108709829258532629170439, 9.228961558712864200498595762894, 10.28856192702944238056431337637, 10.75655375182520575318484283406