L(s) = 1 | + (0.476 + 1.33i)2-s + (0.654 − 0.755i)3-s + (−1.54 + 1.26i)4-s + (0.175 − 1.22i)5-s + (1.31 + 0.511i)6-s + (1.83 + 4.02i)7-s + (−2.42 − 1.45i)8-s + (−0.142 − 0.989i)9-s + (1.71 − 0.349i)10-s + (0.205 + 0.701i)11-s + (−0.0520 + 1.99i)12-s + (1.90 + 0.869i)13-s + (−4.48 + 4.36i)14-s + (−0.809 − 0.934i)15-s + (0.774 − 3.92i)16-s + (2.50 + 3.89i)17-s + ⋯ |
L(s) = 1 | + (0.337 + 0.941i)2-s + (0.378 − 0.436i)3-s + (−0.772 + 0.634i)4-s + (0.0787 − 0.547i)5-s + (0.538 + 0.208i)6-s + (0.694 + 1.52i)7-s + (−0.858 − 0.513i)8-s + (−0.0474 − 0.329i)9-s + (0.541 − 0.110i)10-s + (0.0620 + 0.211i)11-s + (−0.0150 + 0.577i)12-s + (0.528 + 0.241i)13-s + (−1.19 + 1.16i)14-s + (−0.209 − 0.241i)15-s + (0.193 − 0.981i)16-s + (0.606 + 0.943i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.000844 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.000844 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.35156 + 1.35041i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.35156 + 1.35041i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.476 - 1.33i)T \) |
| 3 | \( 1 + (-0.654 + 0.755i)T \) |
| 23 | \( 1 + (2.21 - 4.25i)T \) |
good | 5 | \( 1 + (-0.175 + 1.22i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-1.83 - 4.02i)T + (-4.58 + 5.29i)T^{2} \) |
| 11 | \( 1 + (-0.205 - 0.701i)T + (-9.25 + 5.94i)T^{2} \) |
| 13 | \( 1 + (-1.90 - 0.869i)T + (8.51 + 9.82i)T^{2} \) |
| 17 | \( 1 + (-2.50 - 3.89i)T + (-7.06 + 15.4i)T^{2} \) |
| 19 | \( 1 + (-0.534 + 0.831i)T + (-7.89 - 17.2i)T^{2} \) |
| 29 | \( 1 + (2.81 + 4.38i)T + (-12.0 + 26.3i)T^{2} \) |
| 31 | \( 1 + (1.18 - 1.02i)T + (4.41 - 30.6i)T^{2} \) |
| 37 | \( 1 + (-0.690 - 4.80i)T + (-35.5 + 10.4i)T^{2} \) |
| 41 | \( 1 + (-0.851 + 5.91i)T + (-39.3 - 11.5i)T^{2} \) |
| 43 | \( 1 + (-5.62 - 4.87i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 - 3.20iT - 47T^{2} \) |
| 53 | \( 1 + (4.99 + 10.9i)T + (-34.7 + 40.0i)T^{2} \) |
| 59 | \( 1 + (-3.58 + 7.85i)T + (-38.6 - 44.5i)T^{2} \) |
| 61 | \( 1 + (6.08 + 7.02i)T + (-8.68 + 60.3i)T^{2} \) |
| 67 | \( 1 + (2.30 - 7.86i)T + (-56.3 - 36.2i)T^{2} \) |
| 71 | \( 1 + (-2.19 + 7.48i)T + (-59.7 - 38.3i)T^{2} \) |
| 73 | \( 1 + (11.2 + 7.24i)T + (30.3 + 66.4i)T^{2} \) |
| 79 | \( 1 + (-6.41 + 14.0i)T + (-51.7 - 59.7i)T^{2} \) |
| 83 | \( 1 + (-8.31 + 1.19i)T + (79.6 - 23.3i)T^{2} \) |
| 89 | \( 1 + (11.8 + 10.2i)T + (12.6 + 88.0i)T^{2} \) |
| 97 | \( 1 + (-3.23 - 0.464i)T + (93.0 + 27.3i)T^{2} \) |
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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
−11.31988061405075497715663962488, −9.633505848472215432000699541525, −8.922037140997386799693330178878, −8.283352481452576656131694320219, −7.59549932625553931031475947377, −6.25525089783940231261702925989, −5.62259116363956020376455663683, −4.66461496243610186929099236113, −3.33467224886395039668985321480, −1.78471117076917321401821704873,
1.09564575982117810342372349173, 2.73166898814556962024968439246, 3.76428833797962281862582768497, 4.51922325564586886820437834984, 5.64740792936897523461832370869, 7.04059762361570881217715207948, 8.000721259495888797567837708914, 9.040751870465014490370524918490, 10.01808849071256169668623969457, 10.74975879384979468797876124354