L(s) = 1 | + (2.73 − 1.58i)2-s + (−3.54 − 6.13i)3-s + (0.997 − 1.72i)4-s + 13.6i·5-s + (−19.4 − 11.2i)6-s + (−12.4 − 7.16i)7-s + 18.9i·8-s + (−11.6 + 20.1i)9-s + (21.5 + 37.2i)10-s + (−58.6 + 33.8i)11-s − 14.1·12-s − 45.3·14-s + (83.5 − 48.2i)15-s + (37.9 + 65.7i)16-s + (0.168 − 0.292i)17-s + 73.5i·18-s + ⋯ |
L(s) = 1 | + (0.967 − 0.558i)2-s + (−0.682 − 1.18i)3-s + (0.124 − 0.215i)4-s + 1.21i·5-s + (−1.32 − 0.762i)6-s + (−0.670 − 0.386i)7-s + 0.839i·8-s + (−0.430 + 0.745i)9-s + (0.679 + 1.17i)10-s + (−1.60 + 0.928i)11-s − 0.340·12-s − 0.864·14-s + (1.43 − 0.829i)15-s + (0.593 + 1.02i)16-s + (0.00240 − 0.00417i)17-s + 0.962i·18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000427 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 169 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.000427 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.475348 + 0.475145i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.475348 + 0.475145i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
good | 2 | \( 1 + (-2.73 + 1.58i)T + (4 - 6.92i)T^{2} \) |
| 3 | \( 1 + (3.54 + 6.13i)T + (-13.5 + 23.3i)T^{2} \) |
| 5 | \( 1 - 13.6iT - 125T^{2} \) |
| 7 | \( 1 + (12.4 + 7.16i)T + (171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (58.6 - 33.8i)T + (665.5 - 1.15e3i)T^{2} \) |
| 17 | \( 1 + (-0.168 + 0.292i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-35.0 - 20.2i)T + (3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (77.8 + 134. i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + (-16.8 - 29.2i)T + (-1.21e4 + 2.11e4i)T^{2} \) |
| 31 | \( 1 - 157. iT - 2.97e4T^{2} \) |
| 37 | \( 1 + (50.7 - 29.3i)T + (2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + (51.3 - 29.6i)T + (3.44e4 - 5.96e4i)T^{2} \) |
| 43 | \( 1 + (104. - 180. i)T + (-3.97e4 - 6.88e4i)T^{2} \) |
| 47 | \( 1 + 221. iT - 1.03e5T^{2} \) |
| 53 | \( 1 + 409.T + 1.48e5T^{2} \) |
| 59 | \( 1 + (150. + 86.7i)T + (1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (280. - 485. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-233. + 134. i)T + (1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + (-52.8 - 30.4i)T + (1.78e5 + 3.09e5i)T^{2} \) |
| 73 | \( 1 - 282. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 984.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.20e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + (467. - 269. i)T + (3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + (-1.37e3 - 793. i)T + (4.56e5 + 7.90e5i)T^{2} \) |
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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
−12.63069400911078532046723780693, −11.92147246805893273526592005432, −10.79526693633634944304665242014, −10.16355034218125278153183975629, −7.991161883263513601743431016855, −7.09243243992721683063502418070, −6.21210232773386062090339091860, −4.93264971215540419798124333510, −3.25326628942087544760016843813, −2.19618177261270112676183898133,
0.23157305196068609620612084422, 3.43033810792481412298909373334, 4.69507157471407941699287656090, 5.40068758496163101544705075126, 5.99029226876575462410722833414, 7.86632663000566043494621341992, 9.276510639825065578425324099390, 9.970350735779734734514386709921, 11.11399563109332611298704096367, 12.33855665925114857598263811034