| L(s) = 1 | + (1.65 + 3.42i)2-s + (−2.58 − 3.79i)3-s + (0.946 − 1.18i)4-s + (30.9 − 33.3i)5-s + (8.73 − 15.1i)6-s + (−43.6 + 25.2i)7-s + (64.9 + 14.8i)8-s + (21.8 − 55.7i)9-s + (165. + 51.0i)10-s + (81.5 + 102. i)11-s + (−6.95 − 0.520i)12-s + (−124. + 38.4i)13-s + (−158. − 108. i)14-s + (−206. − 31.1i)15-s + (51.0 + 223. i)16-s + (213. − 198. i)17-s + ⋯ |
| L(s) = 1 | + (0.412 + 0.857i)2-s + (−0.287 − 0.421i)3-s + (0.0591 − 0.0741i)4-s + (1.23 − 1.33i)5-s + (0.242 − 0.420i)6-s + (−0.891 + 0.514i)7-s + (1.01 + 0.231i)8-s + (0.270 − 0.688i)9-s + (1.65 + 0.510i)10-s + (0.673 + 0.844i)11-s + (−0.0482 − 0.00361i)12-s + (−0.737 + 0.227i)13-s + (−0.808 − 0.551i)14-s + (−0.917 − 0.138i)15-s + (0.199 + 0.873i)16-s + (0.740 − 0.686i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.997 + 0.0677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.04202 - 0.0692631i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.04202 - 0.0692631i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-1.84e3 + 170. i)T \) |
| good | 2 | \( 1 + (-1.65 - 3.42i)T + (-9.97 + 12.5i)T^{2} \) |
| 3 | \( 1 + (2.58 + 3.79i)T + (-29.5 + 75.4i)T^{2} \) |
| 5 | \( 1 + (-30.9 + 33.3i)T + (-46.7 - 623. i)T^{2} \) |
| 7 | \( 1 + (43.6 - 25.2i)T + (1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (-81.5 - 102. i)T + (-3.25e3 + 1.42e4i)T^{2} \) |
| 13 | \( 1 + (124. - 38.4i)T + (2.35e4 - 1.60e4i)T^{2} \) |
| 17 | \( 1 + (-213. + 198. i)T + (6.24e3 - 8.32e4i)T^{2} \) |
| 19 | \( 1 + (307. - 120. i)T + (9.55e4 - 8.86e4i)T^{2} \) |
| 23 | \( 1 + (-179. + 27.0i)T + (2.67e5 - 8.24e4i)T^{2} \) |
| 29 | \( 1 + (671. - 984. i)T + (-2.58e5 - 6.58e5i)T^{2} \) |
| 31 | \( 1 + (92.4 - 1.23e3i)T + (-9.13e5 - 1.37e5i)T^{2} \) |
| 37 | \( 1 + (-567. - 327. i)T + (9.37e5 + 1.62e6i)T^{2} \) |
| 41 | \( 1 + (2.22e3 - 1.07e3i)T + (1.76e6 - 2.20e6i)T^{2} \) |
| 47 | \( 1 + (318. - 398. i)T + (-1.08e6 - 4.75e6i)T^{2} \) |
| 53 | \( 1 + (-3.35e3 - 1.03e3i)T + (6.51e6 + 4.44e6i)T^{2} \) |
| 59 | \( 1 + (303. + 1.33e3i)T + (-1.09e7 + 5.25e6i)T^{2} \) |
| 61 | \( 1 + (2.27e3 - 170. i)T + (1.36e7 - 2.06e6i)T^{2} \) |
| 67 | \( 1 + (2.65e3 + 6.77e3i)T + (-1.47e7 + 1.37e7i)T^{2} \) |
| 71 | \( 1 + (-881. + 5.84e3i)T + (-2.42e7 - 7.49e6i)T^{2} \) |
| 73 | \( 1 + (-1.72e3 - 5.58e3i)T + (-2.34e7 + 1.59e7i)T^{2} \) |
| 79 | \( 1 + (-676. - 1.17e3i)T + (-1.94e7 + 3.37e7i)T^{2} \) |
| 83 | \( 1 + (2.79e3 - 1.90e3i)T + (1.73e7 - 4.41e7i)T^{2} \) |
| 89 | \( 1 + (1.02e3 + 1.49e3i)T + (-2.29e7 + 5.84e7i)T^{2} \) |
| 97 | \( 1 + (6.78e3 + 8.50e3i)T + (-1.96e7 + 8.63e7i)T^{2} \) |
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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
−15.18155743773631725460782745930, −14.09160948613312526295948449232, −12.78464455925371054836608542854, −12.29998638172740550443098921851, −9.927844338109545988616988674869, −9.100364915959360087272261174253, −6.98789973381926327854728180308, −6.02775496318864652472459226447, −4.90050859152112167023535874078, −1.54941631574971685790477310473,
2.34031883672106202219706325442, 3.75644160706073468304283054308, 5.94761047485849487062194771541, 7.26822734284136290791252340818, 9.845498483598821109657851584674, 10.44201158328364658840159728646, 11.35829580832785714220564988854, 13.06364759227005953641053919797, 13.70519214106095762163126820933, 14.96916831744248587067898159807
