| L(s) = 1 | + (−5.51 + 11.4i)2-s + (38.3 + 79.6i)3-s + (58.8 + 73.7i)4-s + (121. − 27.6i)5-s − 1.12e3·6-s + 1.96e3i·7-s + (−4.34e3 + 991. i)8-s + (−777. + 975. i)9-s + (−351. + 1.54e3i)10-s + (−1.85e3 + 2.32e3i)11-s + (−3.61e3 + 7.50e3i)12-s + (1.14e4 + 5.03e4i)13-s + (−2.25e4 − 1.08e4i)14-s + (6.84e3 + 8.58e3i)15-s + (7.23e3 − 3.16e4i)16-s + (2.35e4 − 1.03e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.344 + 0.716i)2-s + (0.473 + 0.982i)3-s + (0.229 + 0.288i)4-s + (0.193 − 0.0442i)5-s − 0.866·6-s + 0.818i·7-s + (−1.06 + 0.241i)8-s + (−0.118 + 0.148i)9-s + (−0.0351 + 0.154i)10-s + (−0.126 + 0.159i)11-s + (−0.174 + 0.362i)12-s + (0.402 + 1.76i)13-s + (−0.586 − 0.282i)14-s + (0.135 + 0.169i)15-s + (0.110 − 0.483i)16-s + (0.281 − 1.23i)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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.283162 - 1.69992i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.283162 - 1.69992i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (-3.35e6 - 6.71e5i)T \) |
| good | 2 | \( 1 + (5.51 - 11.4i)T + (-159. - 200. i)T^{2} \) |
| 3 | \( 1 + (-38.3 - 79.6i)T + (-4.09e3 + 5.12e3i)T^{2} \) |
| 5 | \( 1 + (-121. + 27.6i)T + (3.51e5 - 1.69e5i)T^{2} \) |
| 7 | \( 1 - 1.96e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + (1.85e3 - 2.32e3i)T + (-4.76e7 - 2.08e8i)T^{2} \) |
| 13 | \( 1 + (-1.14e4 - 5.03e4i)T + (-7.34e8 + 3.53e8i)T^{2} \) |
| 17 | \( 1 + (-2.35e4 + 1.03e5i)T + (-6.28e9 - 3.02e9i)T^{2} \) |
| 19 | \( 1 + (6.13e4 - 4.89e4i)T + (3.77e9 - 1.65e10i)T^{2} \) |
| 23 | \( 1 + (-1.32e5 + 1.65e5i)T + (-1.74e10 - 7.63e10i)T^{2} \) |
| 29 | \( 1 + (-1.29e5 + 2.69e5i)T + (-3.11e11 - 3.91e11i)T^{2} \) |
| 31 | \( 1 + (1.48e6 + 7.14e5i)T + (5.31e11 + 6.66e11i)T^{2} \) |
| 37 | \( 1 + 2.13e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-2.75e6 - 1.32e6i)T + (4.97e12 + 6.24e12i)T^{2} \) |
| 47 | \( 1 + (-1.93e6 - 2.42e6i)T + (-5.29e12 + 2.32e13i)T^{2} \) |
| 53 | \( 1 + (2.57e6 - 1.12e7i)T + (-5.60e13 - 2.70e13i)T^{2} \) |
| 59 | \( 1 + (2.58e5 - 1.13e6i)T + (-1.32e14 - 6.37e13i)T^{2} \) |
| 61 | \( 1 + (-5.69e6 - 1.18e7i)T + (-1.19e14 + 1.49e14i)T^{2} \) |
| 67 | \( 1 + (1.17e7 + 1.47e7i)T + (-9.03e13 + 3.95e14i)T^{2} \) |
| 71 | \( 1 + (2.35e7 - 1.87e7i)T + (1.43e14 - 6.29e14i)T^{2} \) |
| 73 | \( 1 + (-4.33e6 + 9.89e5i)T + (7.26e14 - 3.49e14i)T^{2} \) |
| 79 | \( 1 + 9.41e4T + 1.51e15T^{2} \) |
| 83 | \( 1 + (3.19e7 - 1.53e7i)T + (1.40e15 - 1.76e15i)T^{2} \) |
| 89 | \( 1 + (-4.11e7 - 8.55e7i)T + (-2.45e15 + 3.07e15i)T^{2} \) |
| 97 | \( 1 + (7.77e7 - 9.74e7i)T + (-1.74e15 - 7.64e15i)T^{2} \) |
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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
−15.09339653715765361535876643461, −14.18863418264708107411710562262, −12.39735871028112895450863788481, −11.23580361014592359028440305668, −9.334362010672229267492106982820, −9.018194352206818327362030742508, −7.36079214982937356638960615000, −5.91422780151193094035771232549, −4.12222411419147813562808338860, −2.46583269609135656714285021172,
0.68091524629526419518902784881, 1.79909383164014931016721692411, 3.28573027867573513572997328756, 5.84912666188576725979508593358, 7.29225464420978223337188797351, 8.496833223780813799793554714740, 10.21960365875538074441621110989, 10.88400859160255012039898248462, 12.56578260684463936675469417826, 13.24876824341803102342428564822
