| L(s) = 1 | + (12.3 + 25.7i)2-s + (−64.7 − 94.9i)3-s + (−348. + 437. i)4-s + (−494. + 532. i)5-s + (1.64e3 − 2.84e3i)6-s + (1.03e3 − 595. i)7-s + (−8.45e3 − 1.92e3i)8-s + (−2.42e3 + 6.18e3i)9-s + (−1.98e4 − 6.11e3i)10-s + (−1.11e4 − 1.40e4i)11-s + (6.41e4 + 4.80e3i)12-s + (3.58e4 − 1.10e4i)13-s + (2.81e4 + 1.91e4i)14-s + (8.26e4 + 1.24e4i)15-s + (−2.32e4 − 1.01e5i)16-s + (6.50e4 − 6.03e4i)17-s + ⋯ |
| L(s) = 1 | + (0.774 + 1.60i)2-s + (−0.799 − 1.17i)3-s + (−1.36 + 1.70i)4-s + (−0.790 + 0.852i)5-s + (1.26 − 2.19i)6-s + (0.429 − 0.248i)7-s + (−2.06 − 0.471i)8-s + (−0.370 + 0.943i)9-s + (−1.98 − 0.611i)10-s + (−0.764 − 0.959i)11-s + (3.09 + 0.231i)12-s + (1.25 − 0.386i)13-s + (0.732 + 0.499i)14-s + (1.63 + 0.245i)15-s + (−0.354 − 1.55i)16-s + (0.778 − 0.722i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.946 + 0.324i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.05137 - 0.175092i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.05137 - 0.175092i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.36e6 - 5.77e5i)T \) |
| good | 2 | \( 1 + (-12.3 - 25.7i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (64.7 + 94.9i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (494. - 532. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (-1.03e3 + 595. i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.11e4 + 1.40e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-3.58e4 + 1.10e4i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (-6.50e4 + 6.03e4i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-5.60e4 + 2.19e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (2.57e5 - 3.87e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (-4.80e5 + 7.05e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (-8.34e4 + 1.11e6i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (3.03e6 + 1.75e6i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (3.32e6 - 1.59e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-4.17e6 + 5.23e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (5.60e6 + 1.73e6i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (2.42e6 + 1.06e7i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-6.04e6 + 4.53e5i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (-8.75e5 - 2.23e6i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-7.19e6 + 4.77e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-3.55e5 - 1.15e6i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (-1.46e7 - 2.53e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-2.21e7 + 1.50e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (2.12e7 + 3.11e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (-2.08e7 - 2.61e7i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.99914470793699099785760488476, −13.43034188231552930279886123204, −12.05373103062803586268709411373, −11.02119072769827005395639282316, −8.085009191501560865684204655693, −7.54470096912831309846055541393, −6.37992618801025076190098673281, −5.47283395634682861497764589913, −3.54992528979158170353278200980, −0.38410721330502717679613910259,
1.39138165150826564841036672705, 3.63875922094671507942200595423, 4.60730647216000747166142039694, 5.41453280424595196629902403645, 8.575101916753432757098760194604, 10.08866358604674013189732811893, 10.78731538009344520537589982410, 11.95781048525373575212306146147, 12.46297104837517167083678704383, 14.02524898829705648510550053735
