| L(s) = 1 | + (−11.8 − 24.5i)2-s + (−24.9 − 36.6i)3-s + (−302. + 378. i)4-s + (−744. + 802. i)5-s + (−603. + 1.04e3i)6-s + (−3.40e3 + 1.96e3i)7-s + (6.06e3 + 1.38e3i)8-s + (1.67e3 − 4.27e3i)9-s + (2.84e4 + 8.78e3i)10-s + (−1.70e4 − 2.14e4i)11-s + (2.14e4 + 1.60e3i)12-s + (−1.59e4 + 4.92e3i)13-s + (8.83e4 + 6.02e4i)14-s + (4.80e4 + 7.23e3i)15-s + (−1.00e4 − 4.42e4i)16-s + (−4.78e3 + 4.44e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.738 − 1.53i)2-s + (−0.308 − 0.452i)3-s + (−1.18 + 1.48i)4-s + (−1.19 + 1.28i)5-s + (−0.465 + 0.806i)6-s + (−1.41 + 0.817i)7-s + (1.48 + 0.338i)8-s + (0.255 − 0.652i)9-s + (2.84 + 0.878i)10-s + (−1.16 − 1.46i)11-s + (1.03 + 0.0774i)12-s + (−0.559 + 0.172i)13-s + (2.29 + 1.56i)14-s + (0.948 + 0.142i)15-s + (−0.153 − 0.674i)16-s + (−0.0572 + 0.0531i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.103 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.182035 - 0.164048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.182035 - 0.164048i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-8.92e5 - 3.30e6i)T \) |
| good | 2 | \( 1 + (11.8 + 24.5i)T + (-159. + 200. i)T^{2} \) |
| 3 | \( 1 + (24.9 + 36.6i)T + (-2.39e3 + 6.10e3i)T^{2} \) |
| 5 | \( 1 + (744. - 802. i)T + (-2.91e4 - 3.89e5i)T^{2} \) |
| 7 | \( 1 + (3.40e3 - 1.96e3i)T + (2.88e6 - 4.99e6i)T^{2} \) |
| 11 | \( 1 + (1.70e4 + 2.14e4i)T + (-4.76e7 + 2.08e8i)T^{2} \) |
| 13 | \( 1 + (1.59e4 - 4.92e3i)T + (6.73e8 - 4.59e8i)T^{2} \) |
| 17 | \( 1 + (4.78e3 - 4.44e3i)T + (5.21e8 - 6.95e9i)T^{2} \) |
| 19 | \( 1 + (-1.22e5 + 4.81e4i)T + (1.24e10 - 1.15e10i)T^{2} \) |
| 23 | \( 1 + (1.00e5 - 1.51e4i)T + (7.48e10 - 2.30e10i)T^{2} \) |
| 29 | \( 1 + (1.04e5 - 1.52e5i)T + (-1.82e11 - 4.65e11i)T^{2} \) |
| 31 | \( 1 + (4.16e4 - 5.55e5i)T + (-8.43e11 - 1.27e11i)T^{2} \) |
| 37 | \( 1 + (-1.84e6 - 1.06e6i)T + (1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (2.88e6 - 1.39e6i)T + (4.97e12 - 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-7.99e5 + 1.00e6i)T + (-5.29e12 - 2.32e13i)T^{2} \) |
| 53 | \( 1 + (8.22e5 + 2.53e5i)T + (5.14e13 + 3.50e13i)T^{2} \) |
| 59 | \( 1 + (5.75e5 + 2.52e6i)T + (-1.32e14 + 6.37e13i)T^{2} \) |
| 61 | \( 1 + (2.45e7 - 1.83e6i)T + (1.89e14 - 2.85e13i)T^{2} \) |
| 67 | \( 1 + (3.09e6 + 7.89e6i)T + (-2.97e14 + 2.76e14i)T^{2} \) |
| 71 | \( 1 + (-5.40e6 + 3.58e7i)T + (-6.17e14 - 1.90e14i)T^{2} \) |
| 73 | \( 1 + (-4.94e6 - 1.60e7i)T + (-6.66e14 + 4.54e14i)T^{2} \) |
| 79 | \( 1 + (8.63e6 + 1.49e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 + (-4.65e7 + 3.17e7i)T + (8.22e14 - 2.09e15i)T^{2} \) |
| 89 | \( 1 + (-3.40e7 - 5.00e7i)T + (-1.43e15 + 3.66e15i)T^{2} \) |
| 97 | \( 1 + (3.57e6 + 4.48e6i)T + (-1.74e15 + 7.64e15i)T^{2} \) |
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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
−13.33303135530244118457484037208, −12.23011291681561818712585158663, −11.54765875892052007578861860145, −10.52132428820371331059811320568, −9.352429958472261485671620896121, −7.85263994913310032024512744830, −6.37103235276913548227477895901, −3.32668051423385040292083497654, −2.88885241858238332691975053148, −0.39637020466478482826274707929,
0.35805175749204977431895104902, 4.31891379232447923542012173158, 5.34425628645361409293017618703, 7.32929191218129406693933843538, 7.79124783067278334153623120587, 9.459123179233756279369042460970, 10.21276223304987095168907976465, 12.36232085857093267972667341976, 13.38976448155929792358359908159, 15.32241149566878608866540510172
