L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−1.61 + 1.03i)11-s + (−0.841 + 0.540i)12-s + (−0.857 − 0.989i)13-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯ |
L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.142 + 0.989i)3-s + (0.415 + 0.909i)4-s + (−0.959 − 0.281i)5-s + (0.415 − 0.909i)6-s + (0.142 − 0.989i)8-s + (−0.959 + 0.281i)9-s + (0.654 + 0.755i)10-s + (−1.61 + 1.03i)11-s + (−0.841 + 0.540i)12-s + (−0.857 − 0.989i)13-s + (0.142 − 0.989i)15-s + (−0.654 + 0.755i)16-s + (0.118 − 0.258i)17-s + (0.959 + 0.281i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.451 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3985116875\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3985116875\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 + (-0.142 - 0.989i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (-0.959 + 0.281i)T \) |
good | 7 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 11 | \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \) |
| 13 | \( 1 + (0.857 + 0.989i)T + (-0.142 + 0.989i)T^{2} \) |
| 17 | \( 1 + (-0.118 + 0.258i)T + (-0.654 - 0.755i)T^{2} \) |
| 19 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 29 | \( 1 + (-0.698 + 1.53i)T + (-0.654 - 0.755i)T^{2} \) |
| 31 | \( 1 + (0.118 - 0.822i)T + (-0.959 - 0.281i)T^{2} \) |
| 37 | \( 1 + (-1.61 + 0.474i)T + (0.841 - 0.540i)T^{2} \) |
| 41 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 43 | \( 1 + (0.186 + 1.29i)T + (-0.959 + 0.281i)T^{2} \) |
| 47 | \( 1 - 0.284T + T^{2} \) |
| 53 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 59 | \( 1 + (-1.25 - 1.45i)T + (-0.142 + 0.989i)T^{2} \) |
| 61 | \( 1 + (0.959 + 0.281i)T^{2} \) |
| 67 | \( 1 + (0.698 + 0.449i)T + (0.415 + 0.909i)T^{2} \) |
| 71 | \( 1 + (-0.415 - 0.909i)T^{2} \) |
| 73 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 79 | \( 1 + (0.544 + 0.627i)T + (-0.142 + 0.989i)T^{2} \) |
| 83 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 89 | \( 1 + (0.959 - 0.281i)T^{2} \) |
| 97 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
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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
−8.911640157077623017690315559402, −8.143862059484581157137976021441, −7.70316140831572799021861832963, −7.03678248010521369881107864032, −5.47371510640362838918640443832, −4.77364938336321150672558540761, −4.06273136983760488882099889995, −2.91033535022226539978346873290, −2.50395829607779896669172978532, −0.39170754691701450919055257679,
1.02271367745274596900914871673, 2.48252876905491602501581738897, 3.08278220692420238046345974599, 4.67597325008722637292716975360, 5.49890031713292977255519739682, 6.41529803205179081143868226211, 7.05854503955512849379069440074, 7.73979378196078867980655116713, 8.132207797142967043724790450564, 8.829198587832584975602466226547