L(s) = 1 | + (−0.160 + 1.40i)2-s + (−1.94 − 0.450i)4-s + (1.93 + 0.705i)5-s + (0.744 + 0.131i)7-s + (0.944 − 2.66i)8-s + (−1.30 + 2.60i)10-s + (0.949 + 2.60i)11-s + (−0.421 − 0.502i)13-s + (−0.303 + 1.02i)14-s + (3.59 + 1.75i)16-s + (3.79 − 2.18i)17-s + (0.155 − 0.270i)19-s + (−3.45 − 2.24i)20-s + (−3.81 + 0.916i)22-s + (1.32 + 7.53i)23-s + ⋯ |
L(s) = 1 | + (−0.113 + 0.993i)2-s + (−0.974 − 0.225i)4-s + (0.866 + 0.315i)5-s + (0.281 + 0.0496i)7-s + (0.333 − 0.942i)8-s + (−0.411 + 0.825i)10-s + (0.286 + 0.786i)11-s + (−0.116 − 0.139i)13-s + (−0.0811 + 0.273i)14-s + (0.898 + 0.438i)16-s + (0.919 − 0.531i)17-s + (0.0357 − 0.0619i)19-s + (−0.773 − 0.502i)20-s + (−0.813 + 0.195i)22-s + (0.276 + 1.57i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.183 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.986727 + 1.18799i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.986727 + 1.18799i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.160 - 1.40i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-1.93 - 0.705i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.744 - 0.131i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.949 - 2.60i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (0.421 + 0.502i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.79 + 2.18i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.155 + 0.270i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.32 - 7.53i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-5.89 - 4.94i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-4.16 + 0.733i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (2.70 - 1.56i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.31 + 5.14i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (3.00 - 1.09i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (1.89 - 10.7i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 0.876T + 53T^{2} \) |
| 59 | \( 1 + (2.94 - 8.09i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-12.1 - 2.14i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-3.32 + 2.79i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (7.08 + 12.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.17 + 12.4i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.09 - 8.44i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-7.10 + 8.47i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (5.55 + 3.20i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.59 + 0.579i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.36460590312392865283054032009, −9.815193938961436316075135679758, −9.112440188096697537731186520878, −8.015593017465574322740039917049, −7.21536064191303600792419562758, −6.39579944666513099360518491344, −5.43897015470285460874812673696, −4.70255368745639947364827545589, −3.23581782368230091303614133580, −1.50747203025334349653778685089,
1.02876377829801302881143774356, 2.27004990026780793908449826304, 3.47561590147970395173565146524, 4.66659753849327743948325047933, 5.56843689540554950849204137329, 6.59990880075746955081996133175, 8.254325691156966598264146405862, 8.541701367919022568934514331076, 9.811441552721868187883899714173, 10.13050209049707691264311382136