L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + (−1.36 + 2.35i)5-s + (−2.13 + 1.56i)7-s + 0.999i·8-s + (−2.35 + 1.36i)10-s + (−0.866 + 0.5i)11-s + 0.193i·13-s + (−2.63 + 0.284i)14-s + (−0.5 + 0.866i)16-s + (−3.07 − 5.32i)17-s + (0.269 + 0.155i)19-s − 2.72·20-s − 0.999·22-s + (−4.50 − 2.60i)23-s + ⋯ |
L(s) = 1 | + (0.612 + 0.353i)2-s + (0.249 + 0.433i)4-s + (−0.608 + 1.05i)5-s + (−0.807 + 0.590i)7-s + 0.353i·8-s + (−0.745 + 0.430i)10-s + (−0.261 + 0.150i)11-s + 0.0537i·13-s + (−0.703 + 0.0759i)14-s + (−0.125 + 0.216i)16-s + (−0.745 − 1.29i)17-s + (0.0617 + 0.0356i)19-s − 0.608·20-s − 0.213·22-s + (−0.939 − 0.542i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.827 + 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6317507105\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6317507105\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 + (2.13 - 1.56i)T \) |
| 11 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (1.36 - 2.35i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 0.193iT - 13T^{2} \) |
| 17 | \( 1 + (3.07 + 5.32i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.269 - 0.155i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.50 + 2.60i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.17iT - 29T^{2} \) |
| 31 | \( 1 + (-1.38 + 0.797i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.21 + 3.82i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.29T + 41T^{2} \) |
| 43 | \( 1 - 4.10T + 43T^{2} \) |
| 47 | \( 1 + (3.63 - 6.29i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.04 + 0.603i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.43 + 9.40i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (8.20 + 4.73i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.69 - 4.66i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 9.63iT - 71T^{2} \) |
| 73 | \( 1 + (2.50 - 1.44i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.60 - 13.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.0T + 83T^{2} \) |
| 89 | \( 1 + (5.67 - 9.83i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 6.77iT - 97T^{2} \) |
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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
−10.05131537964948449985675201599, −9.249359422838197744193151230136, −8.279956169285980833457054981767, −7.34984720106163030757166251846, −6.78574374096112113942449904559, −6.07839319238409056282934583499, −5.05702917448791292624993896365, −4.03277339110081767932653782875, −3.06765211680320165845946986837, −2.43426753300959570412367656448,
0.19796605277603088864165230124, 1.62056941701606854712111139724, 3.08251098264913935514192409701, 4.07906891529301491850942138381, 4.52580408452991927588135770396, 5.74240183043992178982906640655, 6.41636031262311802373473533405, 7.51770995546945237317255674392, 8.305877115021244615199453998690, 9.108489905385948599597990799819