| L(s) = 1 | + (8.92 + 1.12i)3-s + (12.3 + 7.12i)5-s + (11.5 + 20.0i)7-s + (78.4 + 20.0i)9-s + (−2.23 + 1.29i)11-s + (5.00 − 8.67i)13-s + (102. + 77.4i)15-s + 78.0i·17-s + 395.·19-s + (80.8 + 192. i)21-s + (297. + 171. i)23-s + (−211. − 365. i)25-s + (678. + 267. i)27-s + (−1.08e3 + 624. i)29-s + (674. − 1.16e3i)31-s + ⋯ |
| L(s) = 1 | + (0.992 + 0.124i)3-s + (0.493 + 0.284i)5-s + (0.236 + 0.409i)7-s + (0.968 + 0.247i)9-s + (−0.0184 + 0.0106i)11-s + (0.0296 − 0.0513i)13-s + (0.454 + 0.344i)15-s + 0.270i·17-s + 1.09·19-s + (0.183 + 0.435i)21-s + (0.561 + 0.324i)23-s + (−0.337 − 0.584i)25-s + (0.930 + 0.366i)27-s + (−1.28 + 0.742i)29-s + (0.701 − 1.21i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.901 - 0.433i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.40656 + 0.548435i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40656 + 0.548435i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-8.92 - 1.12i)T \) |
| good | 5 | \( 1 + (-12.3 - 7.12i)T + (312.5 + 541. i)T^{2} \) |
| 7 | \( 1 + (-11.5 - 20.0i)T + (-1.20e3 + 2.07e3i)T^{2} \) |
| 11 | \( 1 + (2.23 - 1.29i)T + (7.32e3 - 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-5.00 + 8.67i)T + (-1.42e4 - 2.47e4i)T^{2} \) |
| 17 | \( 1 - 78.0iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 395.T + 1.30e5T^{2} \) |
| 23 | \( 1 + (-297. - 171. i)T + (1.39e5 + 2.42e5i)T^{2} \) |
| 29 | \( 1 + (1.08e3 - 624. i)T + (3.53e5 - 6.12e5i)T^{2} \) |
| 31 | \( 1 + (-674. + 1.16e3i)T + (-4.61e5 - 7.99e5i)T^{2} \) |
| 37 | \( 1 + 1.72e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + (2.74e3 + 1.58e3i)T + (1.41e6 + 2.44e6i)T^{2} \) |
| 43 | \( 1 + (1.20e3 + 2.08e3i)T + (-1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (2.75e3 - 1.58e3i)T + (2.43e6 - 4.22e6i)T^{2} \) |
| 53 | \( 1 + 316. iT - 7.89e6T^{2} \) |
| 59 | \( 1 + (-551. - 318. i)T + (6.05e6 + 1.04e7i)T^{2} \) |
| 61 | \( 1 + (-871. - 1.50e3i)T + (-6.92e6 + 1.19e7i)T^{2} \) |
| 67 | \( 1 + (-3.61e3 + 6.26e3i)T + (-1.00e7 - 1.74e7i)T^{2} \) |
| 71 | \( 1 - 7.09e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 1.75e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + (-902. - 1.56e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (-1.60e3 + 928. i)T + (2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + 3.51e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-7.61e3 - 1.31e4i)T + (-4.42e7 + 7.66e7i)T^{2} \) |
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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
−13.98362297007144040574996698775, −13.18235230984110525475146318896, −11.81567389607173097258637478711, −10.35397103153744752931321555293, −9.383464485299293256107176849672, −8.282418546231945763594324874623, −7.00833850128285663640936687096, −5.30348374384663946128843751517, −3.47570525705962431006701558820, −1.95155386971319558589706432621,
1.51950998994292526584334956945, 3.31303850054422071216938257024, 5.00791818902655186555599863059, 6.87175185985944952187230068912, 8.051027431797016354763767985448, 9.233393249472950060757823964709, 10.15086226703435916275629775240, 11.68688285098518339009076896704, 13.09831444730323843841282064006, 13.73909598605809863122668210090
