| L(s) = 1 | + (0.909 + 1.08i)2-s + (−0.347 + 1.96i)4-s + (−0.696 − 1.91i)5-s + (2.23 + 12.6i)7-s + (−2.44 + 1.41i)8-s + (1.44 − 2.49i)10-s + (−1.28 + 3.52i)11-s + (6.69 + 5.61i)13-s + (−11.6 + 13.9i)14-s + (−3.75 − 1.36i)16-s + (20.4 + 11.8i)17-s + (−2.58 − 4.47i)19-s + (4.01 − 0.707i)20-s + (−4.98 + 1.81i)22-s + (−34.3 − 6.04i)23-s + ⋯ |
| L(s) = 1 | + (0.454 + 0.541i)2-s + (−0.0868 + 0.492i)4-s + (−0.139 − 0.382i)5-s + (0.318 + 1.80i)7-s + (−0.306 + 0.176i)8-s + (0.144 − 0.249i)10-s + (−0.116 + 0.320i)11-s + (0.515 + 0.432i)13-s + (−0.834 + 0.994i)14-s + (−0.234 − 0.0855i)16-s + (1.20 + 0.694i)17-s + (−0.135 − 0.235i)19-s + (0.200 − 0.0353i)20-s + (−0.226 + 0.0824i)22-s + (−1.49 − 0.262i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.100 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.18506 + 1.31023i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.18506 + 1.31023i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.909 - 1.08i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (0.696 + 1.91i)T + (-19.1 + 16.0i)T^{2} \) |
| 7 | \( 1 + (-2.23 - 12.6i)T + (-46.0 + 16.7i)T^{2} \) |
| 11 | \( 1 + (1.28 - 3.52i)T + (-92.6 - 77.7i)T^{2} \) |
| 13 | \( 1 + (-6.69 - 5.61i)T + (29.3 + 166. i)T^{2} \) |
| 17 | \( 1 + (-20.4 - 11.8i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (2.58 + 4.47i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (34.3 + 6.04i)T + (497. + 180. i)T^{2} \) |
| 29 | \( 1 + (18.1 + 21.6i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (0.600 - 3.40i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (-27.7 + 48.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-10.1 + 12.1i)T + (-291. - 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-49.2 - 17.9i)T + (1.41e3 + 1.18e3i)T^{2} \) |
| 47 | \( 1 + (-53.2 + 9.38i)T + (2.07e3 - 755. i)T^{2} \) |
| 53 | \( 1 - 0.286iT - 2.80e3T^{2} \) |
| 59 | \( 1 + (3.64 + 10.0i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (16.4 + 93.1i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-16.2 - 13.6i)T + (779. + 4.42e3i)T^{2} \) |
| 71 | \( 1 + (-89.4 - 51.6i)T + (2.52e3 + 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-11.9 - 20.7i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (9.54 - 8.00i)T + (1.08e3 - 6.14e3i)T^{2} \) |
| 83 | \( 1 + (46.6 + 55.5i)T + (-1.19e3 + 6.78e3i)T^{2} \) |
| 89 | \( 1 + (56.5 - 32.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (31.7 + 11.5i)T + (7.20e3 + 6.04e3i)T^{2} \) |
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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
−12.53166437078265658226192381107, −12.35508155402999853878597294385, −11.18631392213706185102302113132, −9.580095831458691832311292441985, −8.604557989878896799085819638418, −7.83293052409912943684309171991, −6.15411892580812318527757394085, −5.47508331927760724402738731783, −4.10156304849528003620261792302, −2.29960310778749677352127793026,
1.06517228090909460887287716014, 3.26179230315776681015145305324, 4.23558226434556331078932946456, 5.69703980209406968233253473212, 7.13651755504415984700630825743, 8.010331913910326816485354402863, 9.728130010295273892021907158605, 10.58151684085671292783673064522, 11.19653680089472585289850819577, 12.36786558839103018449157399580
