| L(s) = 1 | + (6.51 + 8.17i)2-s + (−88.4 − 13.3i)3-s + (4.15 − 18.2i)4-s + (−356. − 243. i)5-s + (−467. − 810. i)6-s + (−739. + 1.28e3i)7-s + (1.38e3 − 665. i)8-s + (5.55e3 + 1.71e3i)9-s + (−337. − 4.50e3i)10-s + (516. + 2.26e3i)11-s + (−611. + 1.55e3i)12-s + (603. − 8.05e3i)13-s + (−1.52e4 + 2.30e3i)14-s + (2.83e4 + 2.62e4i)15-s + (1.22e4 + 5.91e3i)16-s + (7.53e3 − 5.13e3i)17-s + ⋯ |
| L(s) = 1 | + (0.576 + 0.722i)2-s + (−1.89 − 0.285i)3-s + (0.0324 − 0.142i)4-s + (−1.27 − 0.870i)5-s + (−0.883 − 1.53i)6-s + (−0.814 + 1.41i)7-s + (0.954 − 0.459i)8-s + (2.54 + 0.784i)9-s + (−0.106 − 1.42i)10-s + (0.116 + 0.512i)11-s + (−0.102 + 0.260i)12-s + (0.0762 − 1.01i)13-s + (−1.48 + 0.224i)14-s + (2.16 + 2.01i)15-s + (0.750 + 0.361i)16-s + (0.371 − 0.253i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.671 - 0.740i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.738295 + 0.327232i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.738295 + 0.327232i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-4.09e5 - 3.22e5i)T \) |
| good | 2 | \( 1 + (-6.51 - 8.17i)T + (-28.4 + 124. i)T^{2} \) |
| 3 | \( 1 + (88.4 + 13.3i)T + (2.08e3 + 644. i)T^{2} \) |
| 5 | \( 1 + (356. + 243. i)T + (2.85e4 + 7.27e4i)T^{2} \) |
| 7 | \( 1 + (739. - 1.28e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-516. - 2.26e3i)T + (-1.75e7 + 8.45e6i)T^{2} \) |
| 13 | \( 1 + (-603. + 8.05e3i)T + (-6.20e7 - 9.35e6i)T^{2} \) |
| 17 | \( 1 + (-7.53e3 + 5.13e3i)T + (1.49e8 - 3.81e8i)T^{2} \) |
| 19 | \( 1 + (1.28e4 - 3.96e3i)T + (7.38e8 - 5.03e8i)T^{2} \) |
| 23 | \( 1 + (2.63e4 - 2.44e4i)T + (2.54e8 - 3.39e9i)T^{2} \) |
| 29 | \( 1 + (-1.80e5 + 2.72e4i)T + (1.64e10 - 5.08e9i)T^{2} \) |
| 31 | \( 1 + (-4.99e3 + 1.27e4i)T + (-2.01e10 - 1.87e10i)T^{2} \) |
| 37 | \( 1 + (-4.26e4 - 7.39e4i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + (-4.21e5 - 5.28e5i)T + (-4.33e10 + 1.89e11i)T^{2} \) |
| 47 | \( 1 + (7.45e4 - 3.26e5i)T + (-4.56e11 - 2.19e11i)T^{2} \) |
| 53 | \( 1 + (4.16e4 + 5.55e5i)T + (-1.16e12 + 1.75e11i)T^{2} \) |
| 59 | \( 1 + (-4.88e3 - 2.35e3i)T + (1.55e12 + 1.94e12i)T^{2} \) |
| 61 | \( 1 + (-2.40e5 - 6.13e5i)T + (-2.30e12 + 2.13e12i)T^{2} \) |
| 67 | \( 1 + (-1.67e6 + 5.15e5i)T + (5.00e12 - 3.41e12i)T^{2} \) |
| 71 | \( 1 + (-3.06e5 - 2.84e5i)T + (6.79e11 + 9.06e12i)T^{2} \) |
| 73 | \( 1 + (2.37e5 - 3.17e6i)T + (-1.09e13 - 1.64e12i)T^{2} \) |
| 79 | \( 1 + (-3.99e4 + 6.92e4i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.19e6 - 1.79e5i)T + (2.59e13 + 7.99e12i)T^{2} \) |
| 89 | \( 1 + (-1.01e6 - 1.53e5i)T + (4.22e13 + 1.30e13i)T^{2} \) |
| 97 | \( 1 + (1.04e6 + 4.57e6i)T + (-7.27e13 + 3.50e13i)T^{2} \) |
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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
−15.23296643301505648719606738931, −12.83552690174916167063714362081, −12.42624965613433694659376940619, −11.44218383598557551510241763849, −9.949763974008036447495958220317, −7.81488512718466015960439280888, −6.39343259272927905713055778124, −5.50188106844143305518111933707, −4.50762188746168454500850839204, −0.817859463876698765218203828785,
0.56984415696748579578270357594, 3.71806896918596272456897326274, 4.34707608274485457861240372300, 6.51403859105389507792462211705, 7.36653541771339739074596997170, 10.40363958558851785599114155668, 10.88448881547595154716977972635, 11.78513302810183237727069532928, 12.57889235521878895613330987012, 14.01462629073526571883847590778
