| L(s) = 1 | + (−1.27 + 0.612i)2-s + (−1.51 − 0.842i)3-s + (1.24 − 1.56i)4-s + (−1.15 + 3.16i)5-s + (2.44 + 0.146i)6-s + (0.984 + 0.173i)7-s + (−0.636 + 2.75i)8-s + (1.58 + 2.54i)9-s + (−0.469 − 4.73i)10-s + (2.52 − 0.919i)11-s + (−3.20 + 1.31i)12-s + (1.75 − 1.47i)13-s + (−1.36 + 0.381i)14-s + (4.40 − 3.81i)15-s + (−0.876 − 3.90i)16-s + (−4.09 + 2.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.901 + 0.433i)2-s + (−0.873 − 0.486i)3-s + (0.624 − 0.780i)4-s + (−0.514 + 1.41i)5-s + (0.998 + 0.0596i)6-s + (0.372 + 0.0656i)7-s + (−0.225 + 0.974i)8-s + (0.527 + 0.849i)9-s + (−0.148 − 1.49i)10-s + (0.761 − 0.277i)11-s + (−0.925 + 0.378i)12-s + (0.485 − 0.407i)13-s + (−0.363 + 0.102i)14-s + (1.13 − 0.985i)15-s + (−0.219 − 0.975i)16-s + (−0.992 + 0.572i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.511 - 0.859i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.280886 + 0.493787i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.280886 + 0.493787i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.27 - 0.612i)T \) |
| 3 | \( 1 + (1.51 + 0.842i)T \) |
| 7 | \( 1 + (-0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (1.15 - 3.16i)T + (-3.83 - 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.52 + 0.919i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.75 + 1.47i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (4.09 - 2.36i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.50 - 2.02i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.542 + 3.07i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.207 - 0.247i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (2.34 - 0.414i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (1.29 + 2.23i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-5.06 - 6.04i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (-3.74 - 10.2i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.41 - 8.03i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 2.58iT - 53T^{2} \) |
| 59 | \( 1 + (-1.37 - 0.499i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.54 - 14.4i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (8.59 + 10.2i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-1.13 - 1.96i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (7.35 - 12.7i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.624 - 0.744i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (6.80 + 5.70i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (13.1 + 7.57i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.09 - 2.21i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.80925769225386642935290352194, −9.973971731334730224718769153108, −8.768221757641884133689385833200, −7.78935248123785626558940775892, −7.22824688318687498625050370688, −6.30598443669091100793301365319, −5.90468601691866033287634782217, −4.35904287822772171696302579354, −2.79322811797251721290172496837, −1.34465030497222073399813499432,
0.49266289057024574840025117063, 1.64687515983210528783443386911, 3.73196114467243280029934130794, 4.45793033166370348018758305557, 5.44947489312630828025401283157, 6.75004625357838225616107893098, 7.54115528058579950800059202749, 8.902994287539328286273989393141, 8.992214708025156074384125347995, 9.963425299874498567018753434079
