| 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 | \( 1 + (-3.21 - 1.17i)T + (74.3 + 62.3i)T^{2} \) |
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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
−12.56618096726534642961249192815, −12.09531728767560225765001301928, −10.65232552576651475077018283429, −9.930504450018799381150212330491, −8.857832673936826461871289435410, −7.35556027597914959278157158464, −6.06229404910769250481742251159, −4.99397791544914048359135871678, −3.58640312145986509602051705520, −1.85363419930606385773560585482,
2.45927333243607376973869572282, 4.18519048863507980954315565373, 5.40410070804570242472191486453, 6.66085577282787644193434851316, 7.40110024478664495350077199924, 9.117748828551849507147723348007, 9.646832305796472282128423008494, 11.47122738211707393808910922789, 11.82023607638582893640025007046, 13.39058495953011650579970805847
