L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 − 0.5i)3-s + (0.499 − 0.866i)4-s + (0.133 + 2.23i)5-s + 0.999·6-s + 0.999i·8-s + (0.499 + 0.866i)9-s + (−1.23 − 1.86i)10-s + (−1 + 1.73i)11-s + (−0.866 + 0.499i)12-s − 6i·13-s + (1 − 1.99i)15-s + (−0.5 − 0.866i)16-s + (−1.73 − i)17-s + (−0.866 − 0.499i)18-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.499 − 0.288i)3-s + (0.249 − 0.433i)4-s + (0.0599 + 0.998i)5-s + 0.408·6-s + 0.353i·8-s + (0.166 + 0.288i)9-s + (−0.389 − 0.590i)10-s + (−0.301 + 0.522i)11-s + (−0.249 + 0.144i)12-s − 1.66i·13-s + (0.258 − 0.516i)15-s + (−0.125 − 0.216i)16-s + (−0.420 − 0.242i)17-s + (−0.204 − 0.117i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8708326706\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8708326706\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 5 | \( 1 + (-0.133 - 2.23i)T \) |
| 7 | \( 1 \) |
good | 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 6iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 + i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + (-4 + 6.92i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.73 + i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 + (6.92 - 4i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-5.19 - 3i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1 + 1.73i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.92 - 4i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (3.46 + 2i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + (5 + 8.66i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−9.782904012736273616980722425805, −8.442323891463334527925734846472, −7.73406528286800490707871481343, −7.12492774752850324512130069040, −6.29965613699331467012539237489, −5.61965115653977744326222805700, −4.61656698214323637008634901469, −3.12026566243053883354757650983, −2.22808566479561012462342435614, −0.57888197889722498363021273807,
0.974165179803274292867103777158, 2.10481797138180298725424723454, 3.59036905987497069791817264788, 4.51442523509593169177359648808, 5.28256237295560185596622820414, 6.37391262335333477219694472631, 7.11042891740407071941076229614, 8.289863268740059580271871386859, 8.857237781694999914596945519629, 9.464231018467644353391284848187