| L(s) = 1 | + (0.766 + 0.642i)2-s + (0.173 + 0.984i)4-s + (1.26 + 0.460i)5-s + (−0.0209 + 0.118i)7-s + (−0.500 + 0.866i)8-s + (0.673 + 1.16i)10-s + (3.49 − 1.27i)11-s + (−4.64 + 3.89i)13-s + (−0.0923 + 0.0775i)14-s + (−0.939 + 0.342i)16-s + (−2.58 − 4.47i)17-s + (2.96 − 5.12i)19-s + (−0.233 + 1.32i)20-s + (3.49 + 1.27i)22-s + (0.826 + 4.68i)23-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.0868 + 0.492i)4-s + (0.566 + 0.206i)5-s + (−0.00791 + 0.0448i)7-s + (−0.176 + 0.306i)8-s + (0.213 + 0.368i)10-s + (1.05 − 0.383i)11-s + (−1.28 + 1.08i)13-s + (−0.0246 + 0.0207i)14-s + (−0.234 + 0.0855i)16-s + (−0.626 − 1.08i)17-s + (0.679 − 1.17i)19-s + (−0.0523 + 0.296i)20-s + (0.744 + 0.271i)22-s + (0.172 + 0.977i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.43624 + 0.619537i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43624 + 0.619537i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.766 - 0.642i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-1.26 - 0.460i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.0209 - 0.118i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-3.49 + 1.27i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (4.64 - 3.89i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.58 + 4.47i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.96 + 5.12i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.826 - 4.68i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (4.55 + 3.82i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.875 + 4.96i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (0.145 + 0.251i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.44 - 3.72i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.426 - 0.155i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.134 + 0.761i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 7.29T + 53T^{2} \) |
| 59 | \( 1 + (1.40 + 0.509i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (0.656 - 3.72i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (5.08 - 4.26i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-2.87 - 4.97i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (5.20 - 9.02i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.7 - 8.99i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-1.81 - 1.52i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (1.08 - 1.87i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.21 + 1.17i)T + (74.3 - 62.3i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.39058495953011650579970805847, −11.82023607638582893640025007046, −11.47122738211707393808910922789, −9.646832305796472282128423008494, −9.117748828551849507147723348007, −7.40110024478664495350077199924, −6.66085577282787644193434851316, −5.40410070804570242472191486453, −4.18519048863507980954315565373, −2.45927333243607376973869572282,
1.85363419930606385773560585482, 3.58640312145986509602051705520, 4.99397791544914048359135871678, 6.06229404910769250481742251159, 7.35556027597914959278157158464, 8.857832673936826461871289435410, 9.930504450018799381150212330491, 10.65232552576651475077018283429, 12.09531728767560225765001301928, 12.56618096726534642961249192815
