| L(s) = 1 | + 2i·3-s + (−3 + 3i)7-s − 9-s + (2 + 2i)11-s + (−2 + 2i)13-s + 6·17-s + (−4 + 4i)19-s + (−6 − 6i)21-s + (5 + 5i)23-s + 4i·27-s + (−2 + 5i)29-s + (−6 − 6i)31-s + (−4 + 4i)33-s − 6i·37-s + (−4 − 4i)39-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + (−1.13 + 1.13i)7-s − 0.333·9-s + (0.603 + 0.603i)11-s + (−0.554 + 0.554i)13-s + 1.45·17-s + (−0.917 + 0.917i)19-s + (−1.30 − 1.30i)21-s + (1.04 + 1.04i)23-s + 0.769i·27-s + (−0.371 + 0.928i)29-s + (−1.07 − 1.07i)31-s + (−0.696 + 0.696i)33-s − 0.986i·37-s + (−0.640 − 0.640i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.252671274\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.252671274\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 29 | \( 1 + (2 - 5i)T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 7 | \( 1 + (3 - 3i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2 - 2i)T + 11iT^{2} \) |
| 13 | \( 1 + (2 - 2i)T - 13iT^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + (4 - 4i)T - 19iT^{2} \) |
| 23 | \( 1 + (-5 - 5i)T + 23iT^{2} \) |
| 31 | \( 1 + (6 + 6i)T + 31iT^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + (-7 + 7i)T - 41iT^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8iT - 47T^{2} \) |
| 53 | \( 1 + (-4 - 4i)T + 53iT^{2} \) |
| 59 | \( 1 + 4iT - 59T^{2} \) |
| 61 | \( 1 + (-9 - 9i)T + 61iT^{2} \) |
| 67 | \( 1 + (3 + 3i)T + 67iT^{2} \) |
| 71 | \( 1 + 12iT - 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + (8 - 8i)T - 79iT^{2} \) |
| 83 | \( 1 + (-3 - 3i)T + 83iT^{2} \) |
| 89 | \( 1 + (-7 + 7i)T - 89iT^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−9.379768314640414115649867775725, −8.880613867348480001755859510985, −7.58515957344434026104251885667, −6.96433189869470152212222528860, −5.76657296274509949119021804799, −5.54661094904062123309254949361, −4.34333822966742514393820838722, −3.70798527963776588288731338273, −2.90102209465193200658671914853, −1.71678062882330372850353226323,
0.44204649216281090613912552256, 1.14984765122851659209047568091, 2.58166436293935809995596778845, 3.38524268383881273635851517533, 4.27967073720307300271273431835, 5.43582126961429306085295709244, 6.39807696666865248092829321780, 6.86538486156140705656128655942, 7.37921140321126875248277844210, 8.193529009059072443594106182959
