L(s) = 1 | − 4.45·2-s + (9.81 − 16.9i)3-s − 12.1·4-s + (−39.8 + 69.0i)5-s + (−43.6 + 75.6i)6-s + (8.78 + 15.2i)7-s + 196.·8-s + (−71.0 − 123. i)9-s + (177. − 307. i)10-s + 296.·11-s + (−119. + 207. i)12-s + (198. + 343. i)13-s + (−39.1 − 67.7i)14-s + (782. + 1.35e3i)15-s − 485.·16-s + (676. + 1.17e3i)17-s + ⋯ |
L(s) = 1 | − 0.786·2-s + (0.629 − 1.09i)3-s − 0.380·4-s + (−0.713 + 1.23i)5-s + (−0.495 + 0.857i)6-s + (0.0677 + 0.117i)7-s + 1.08·8-s + (−0.292 − 0.506i)9-s + (0.561 − 0.972i)10-s + 0.738·11-s + (−0.239 + 0.415i)12-s + (0.325 + 0.563i)13-s + (−0.0533 − 0.0923i)14-s + (0.898 + 1.55i)15-s − 0.474·16-s + (0.567 + 0.982i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.621 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.831073 + 0.401426i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.831073 + 0.401426i\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 + (-865. - 1.20e4i)T \) |
good | 2 | \( 1 + 4.45T + 32T^{2} \) |
| 3 | \( 1 + (-9.81 + 16.9i)T + (-121.5 - 210. i)T^{2} \) |
| 5 | \( 1 + (39.8 - 69.0i)T + (-1.56e3 - 2.70e3i)T^{2} \) |
| 7 | \( 1 + (-8.78 - 15.2i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 - 296.T + 1.61e5T^{2} \) |
| 13 | \( 1 + (-198. - 343. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 + (-676. - 1.17e3i)T + (-7.09e5 + 1.22e6i)T^{2} \) |
| 19 | \( 1 + (680. - 1.17e3i)T + (-1.23e6 - 2.14e6i)T^{2} \) |
| 23 | \( 1 + (570. - 987. i)T + (-3.21e6 - 5.57e6i)T^{2} \) |
| 29 | \( 1 + (1.19e3 + 2.07e3i)T + (-1.02e7 + 1.77e7i)T^{2} \) |
| 31 | \( 1 + (2.83e3 - 4.91e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (3.03e3 - 5.26e3i)T + (-3.46e7 - 6.00e7i)T^{2} \) |
| 41 | \( 1 - 1.04e4T + 1.15e8T^{2} \) |
| 47 | \( 1 + 2.68e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + (-1.51e4 + 2.62e4i)T + (-2.09e8 - 3.62e8i)T^{2} \) |
| 59 | \( 1 + 3.00e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + (6.41e3 + 1.11e4i)T + (-4.22e8 + 7.31e8i)T^{2} \) |
| 67 | \( 1 + (1.94e4 - 3.37e4i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-2.31e4 - 4.01e4i)T + (-9.02e8 + 1.56e9i)T^{2} \) |
| 73 | \( 1 + (-1.48e4 - 2.57e4i)T + (-1.03e9 + 1.79e9i)T^{2} \) |
| 79 | \( 1 + (-3.40e4 - 5.89e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-3.83e4 + 6.64e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + (-6.45e4 + 1.11e5i)T + (-2.79e9 - 4.83e9i)T^{2} \) |
| 97 | \( 1 + 5.70e4T + 8.58e9T^{2} \) |
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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
−14.74232612988920559545054143902, −14.12884878654671909548817299140, −12.84414654224267357240513158164, −11.44678855004943839230644529932, −10.13614138587058482110914705761, −8.564075538766958881318868029931, −7.70876862262568875575860493283, −6.62373306572772777144431686258, −3.71201309120114240770580918673, −1.62549367191514006370273365810,
0.67875054881750469881790122093, 3.88437045429982784314069203981, 4.88248304534816697803091476242, 7.76997190502498829849944331772, 8.915373714387469585941932100035, 9.375901133675999276786151866067, 10.77859529314794697274423243539, 12.38415923541474853569794091063, 13.74168541795648646105243028118, 15.02304194038131780092239215529