L(s) = 1 | + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (0.195 + 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 − 0.555i)18-s + (−0.707 + 0.292i)19-s + (0.831 + 0.555i)20-s + ⋯ |
L(s) = 1 | + (−0.195 + 0.980i)2-s + (−0.555 − 0.831i)3-s + (−0.923 − 0.382i)4-s + (−0.980 − 0.195i)5-s + (0.923 − 0.382i)6-s + (0.555 − 0.831i)8-s + (−0.382 + 0.923i)9-s + (0.382 − 0.923i)10-s + (0.195 + 0.980i)12-s + (0.382 + 0.923i)15-s + (0.707 + 0.707i)16-s + (−0.831 + 0.555i)17-s + (−0.831 − 0.555i)18-s + (−0.707 + 0.292i)19-s + (0.831 + 0.555i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.939 - 0.341i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2007522275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2007522275\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.195 - 0.980i)T \) |
| 3 | \( 1 + (0.555 + 0.831i)T \) |
| 5 | \( 1 + (0.980 + 0.195i)T \) |
| 17 | \( 1 + (0.831 - 0.555i)T \) |
good | 7 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 11 | \( 1 + (-0.382 - 0.923i)T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 19 | \( 1 + (0.707 - 0.292i)T + (0.707 - 0.707i)T^{2} \) |
| 23 | \( 1 + (0.785 - 1.17i)T + (-0.382 - 0.923i)T^{2} \) |
| 29 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 31 | \( 1 + (1.63 - 1.08i)T + (0.382 - 0.923i)T^{2} \) |
| 37 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 41 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 43 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + (0.275 - 0.275i)T - iT^{2} \) |
| 53 | \( 1 + (-0.425 - 1.02i)T + (-0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.382 - 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 79 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 83 | \( 1 + (-0.636 - 1.53i)T + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 + iT^{2} \) |
| 97 | \( 1 + (0.923 + 0.382i)T^{2} \) |
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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
−10.69116283257771374655000852561, −9.420171794958841637113316146729, −8.491788441920866261458490176961, −7.921891124314710482105688596626, −7.18238072665353756646505448809, −6.45259923837375032015508659923, −5.56600050359970285189920591623, −4.65075396149465432620644008841, −3.64530506494762668126937678153, −1.62994738404096938322916247199,
0.21669656063834619983073197022, 2.40579435070972329083106504992, 3.61379322224330567630790825518, 4.28361741139901757862588359760, 5.01895012307769842000564684276, 6.30127934533145608779108588647, 7.41259406526613279962414300237, 8.498164230137040473550419847299, 9.046850948507717696220462979967, 10.02454124336541756594670031459