| L(s) = 1 | − 17.3·2-s + (−16.5 − 28.5i)3-s + 172.·4-s + (−197. − 342. i)5-s + (286. + 495. i)6-s + (401. − 694. i)7-s − 767.·8-s + (548. − 949. i)9-s + (3.42e3 + 5.93e3i)10-s + 7.43e3·11-s + (−2.84e3 − 4.92e3i)12-s + (3.63e3 − 6.29e3i)13-s + (−6.95e3 + 1.20e4i)14-s + (−6.53e3 + 1.13e4i)15-s − 8.75e3·16-s + (1.35e4 − 2.34e4i)17-s + ⋯ |
| L(s) = 1 | − 1.53·2-s + (−0.353 − 0.611i)3-s + 1.34·4-s + (−0.707 − 1.22i)5-s + (0.540 + 0.936i)6-s + (0.442 − 0.765i)7-s − 0.530·8-s + (0.250 − 0.434i)9-s + (1.08 + 1.87i)10-s + 1.68·11-s + (−0.475 − 0.823i)12-s + (0.458 − 0.794i)13-s + (−0.677 + 1.17i)14-s + (−0.499 + 0.865i)15-s − 0.534·16-s + (0.666 − 1.15i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.113432 - 0.753387i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.113432 - 0.753387i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (4.70e5 + 2.23e5i)T \) |
| good | 2 | \( 1 + 17.3T + 128T^{2} \) |
| 3 | \( 1 + (16.5 + 28.5i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (197. + 342. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-401. + 694. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 - 7.43e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (-3.63e3 + 6.29e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-1.35e4 + 2.34e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (3.18e3 + 5.50e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (2.87e4 + 4.97e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-1.09e5 + 1.89e5i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-1.14e5 - 1.98e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-1.68e5 - 2.92e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 - 1.27e5T + 1.94e11T^{2} \) |
| 47 | \( 1 - 6.96e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (-9.57e4 - 1.65e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 - 1.31e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (2.87e5 - 4.97e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (-1.00e6 - 1.74e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (8.46e5 - 1.46e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.39e6 + 4.14e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (3.70e6 - 6.41e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-2.85e6 - 4.95e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-9.86e4 - 1.70e5i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.41e6T + 8.07e13T^{2} \) |
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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
−13.81608629141779669032245395644, −12.22695597596576754368655130795, −11.56116537705972883595555217121, −9.979175064259605592269290477862, −8.774947937982920738676839195465, −7.83992032949814190417342149967, −6.65454575420594858535747052688, −4.32604977877250737036973342425, −0.966814811935267700538925470269, −0.852219204251698748996480896687,
1.68987960796299677861149536288, 3.95613744750818381468492846124, 6.36340310066171449628913012743, 7.66759826724628518845446692292, 8.910247048506542609707978226004, 10.11164780547688431551259497021, 11.14021168091943002679230897287, 11.77829287245880498826508841340, 14.30837581491281295873914060763, 15.28843486652117073426690605728
