| 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
−15.02304194038131780092239215529, −13.74168541795648646105243028118, −12.38415923541474853569794091063, −10.77859529314794697274423243539, −9.375901133675999276786151866067, −8.915373714387469585941932100035, −7.76997190502498829849944331772, −4.88248304534816697803091476242, −3.88437045429982784314069203981, −0.67875054881750469881790122093,
1.62549367191514006370273365810, 3.71201309120114240770580918673, 6.62373306572772777144431686258, 7.70876862262568875575860493283, 8.564075538766958881318868029931, 10.13614138587058482110914705761, 11.44678855004943839230644529932, 12.84414654224267357240513158164, 14.12884878654671909548817299140, 14.74232612988920559545054143902
