L(s) = 1 | + (−0.5 − 0.866i)2-s + (−0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s + (−0.5 + 0.866i)5-s + 0.999·6-s + (2.09 − 3.62i)7-s + 0.999·8-s + (−0.499 − 0.866i)9-s + 0.999·10-s − 5.42·11-s + (−0.499 − 0.866i)12-s + (2.00 − 3.46i)13-s − 4.18·14-s + (−0.499 − 0.866i)15-s + (−0.5 − 0.866i)16-s + (2.86 + 4.96i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s + (−0.223 + 0.387i)5-s + 0.408·6-s + (0.790 − 1.36i)7-s + 0.353·8-s + (−0.166 − 0.288i)9-s + 0.316·10-s − 1.63·11-s + (−0.144 − 0.249i)12-s + (0.555 − 0.961i)13-s − 1.11·14-s + (−0.129 − 0.223i)15-s + (−0.125 − 0.216i)16-s + (0.694 + 1.20i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.967 + 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4090147672\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4090147672\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.5 - 0.866i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 37 | \( 1 + (4.85 + 3.66i)T \) |
good | 7 | \( 1 + (-2.09 + 3.62i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + 5.42T + 11T^{2} \) |
| 13 | \( 1 + (-2.00 + 3.46i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.86 - 4.96i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.631 - 1.09i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 5.55T + 23T^{2} \) |
| 29 | \( 1 - 0.557T + 29T^{2} \) |
| 31 | \( 1 + 5.22T + 31T^{2} \) |
| 41 | \( 1 + (-0.382 + 0.662i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 - 9.95T + 47T^{2} \) |
| 53 | \( 1 + (2.62 + 4.54i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.76 + 11.7i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-7.04 + 12.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.69 - 4.66i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.70 - 6.42i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + 16.0T + 73T^{2} \) |
| 79 | \( 1 + (-3.35 + 5.81i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.71 + 2.97i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (7.45 + 12.9i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 0.137T + 97T^{2} \) |
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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
−10.02890702196778401531785815536, −8.430983653631210690088084954657, −7.977811555023334861975424705905, −7.30724233319358291132837418694, −5.89728276903165426295718557322, −5.02021667888574954522630912138, −3.93462540533175984843476062906, −3.31799258226891378768506310736, −1.77336913306248169021338905229, −0.20713576853572433885579660203,
1.63333789726635098541674284946, 2.71932402457632544111073779366, 4.53782709780193027060834327428, 5.39530161158130471373403873214, 5.79840446432145144048795631965, 7.06249843369903257756262568476, 7.80769685098220621202171238100, 8.486064667156162274494903149543, 9.075125455219823858857293640205, 10.12561899289402140309080312689