| L(s) = 1 | + (−2.01 − 8.84i)2-s + (52.4 + 48.6i)3-s + (387. − 186. i)4-s + (1.70e3 − 257. i)5-s + (324. − 562. i)6-s + (−3.89e3 − 6.75e3i)7-s + (−5.32e3 − 6.67e3i)8-s + (−1.08e3 − 1.45e4i)9-s + (−5.72e3 − 1.45e4i)10-s + (2.24e4 + 1.08e4i)11-s + (2.93e4 + 9.06e3i)12-s + (−2.46e4 + 6.27e4i)13-s + (−5.18e4 + 4.81e4i)14-s + (1.02e5 + 6.96e4i)15-s + (8.88e4 − 1.11e5i)16-s + (−1.42e5 − 2.15e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.0892 − 0.390i)2-s + (0.374 + 0.347i)3-s + (0.756 − 0.364i)4-s + (1.22 − 0.184i)5-s + (0.102 − 0.177i)6-s + (−0.613 − 1.06i)7-s + (−0.459 − 0.576i)8-s + (−0.0552 − 0.737i)9-s + (−0.180 − 0.461i)10-s + (0.462 + 0.222i)11-s + (0.409 + 0.126i)12-s + (−0.239 + 0.609i)13-s + (−0.360 + 0.334i)14-s + (0.521 + 0.355i)15-s + (0.338 − 0.425i)16-s + (−0.414 − 0.0624i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0609 + 0.998i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.0609 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.88330 - 2.00184i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.88330 - 2.00184i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-2.23e7 + 1.29e6i)T \) |
| good | 2 | \( 1 + (2.01 + 8.84i)T + (-461. + 222. i)T^{2} \) |
| 3 | \( 1 + (-52.4 - 48.6i)T + (1.47e3 + 1.96e4i)T^{2} \) |
| 5 | \( 1 + (-1.70e3 + 257. i)T + (1.86e6 - 5.75e5i)T^{2} \) |
| 7 | \( 1 + (3.89e3 + 6.75e3i)T + (-2.01e7 + 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.24e4 - 1.08e4i)T + (1.47e9 + 1.84e9i)T^{2} \) |
| 13 | \( 1 + (2.46e4 - 6.27e4i)T + (-7.77e9 - 7.21e9i)T^{2} \) |
| 17 | \( 1 + (1.42e5 + 2.15e4i)T + (1.13e11 + 3.49e10i)T^{2} \) |
| 19 | \( 1 + (-1.26e4 + 1.68e5i)T + (-3.19e11 - 4.80e10i)T^{2} \) |
| 23 | \( 1 + (6.96e5 - 4.74e5i)T + (6.58e11 - 1.67e12i)T^{2} \) |
| 29 | \( 1 + (7.69e4 - 7.13e4i)T + (1.08e12 - 1.44e13i)T^{2} \) |
| 31 | \( 1 + (7.33e5 + 2.26e5i)T + (2.18e13 + 1.48e13i)T^{2} \) |
| 37 | \( 1 + (-1.09e7 + 1.89e7i)T + (-6.49e13 - 1.12e14i)T^{2} \) |
| 41 | \( 1 + (-4.87e6 - 2.13e7i)T + (-2.94e14 + 1.42e14i)T^{2} \) |
| 47 | \( 1 + (-1.31e7 + 6.34e6i)T + (6.97e14 - 8.74e14i)T^{2} \) |
| 53 | \( 1 + (-4.15e6 - 1.05e7i)T + (-2.41e15 + 2.24e15i)T^{2} \) |
| 59 | \( 1 + (1.16e5 - 1.46e5i)T + (-1.92e15 - 8.44e15i)T^{2} \) |
| 61 | \( 1 + (5.35e6 - 1.65e6i)T + (9.66e15 - 6.58e15i)T^{2} \) |
| 67 | \( 1 + (-4.27e5 + 5.70e6i)T + (-2.69e16 - 4.05e15i)T^{2} \) |
| 71 | \( 1 + (-2.75e8 - 1.88e8i)T + (1.67e16 + 4.26e16i)T^{2} \) |
| 73 | \( 1 + (8.04e6 - 2.05e7i)T + (-4.31e16 - 4.00e16i)T^{2} \) |
| 79 | \( 1 + (-3.03e8 - 5.26e8i)T + (-5.99e16 + 1.03e17i)T^{2} \) |
| 83 | \( 1 + (-1.27e8 - 1.17e8i)T + (1.39e16 + 1.86e17i)T^{2} \) |
| 89 | \( 1 + (-3.27e7 - 3.03e7i)T + (2.61e16 + 3.49e17i)T^{2} \) |
| 97 | \( 1 + (7.81e8 + 3.76e8i)T + (4.73e17 + 5.94e17i)T^{2} \) |
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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
−13.75966805935519730914145722892, −12.49736627984483156679968505167, −11.05458854258011701706467904515, −9.780648634794051398660324047892, −9.398108270934186171949724519410, −7.01061943102966649979420415439, −6.04512876693946126520144536305, −3.92822897355528289683811405260, −2.34070910645852146995877370351, −0.914282423433271740926402612516,
2.00413710291308981870220061443, 2.81542072056887628775541709088, 5.64109977044492601648823009340, 6.49499013748685722159410320609, 7.988533979758559621744276432280, 9.199999531278150990149605093370, 10.59135747017655850944350220797, 12.10302565317314450461307979646, 13.16247817827915660499333610808, 14.30145263104355337430774041032
