| 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
−10.74975879384979468797876124354, −10.01808849071256169668623969457, −9.040751870465014490370524918490, −8.000721259495888797567837708914, −7.04059762361570881217715207948, −5.64740792936897523461832370869, −4.51922325564586886820437834984, −3.76428833797962281862582768497, −2.73166898814556962024968439246, −1.09564575982117810342372349173,
1.78471117076917321401821704873, 3.33467224886395039668985321480, 4.66461496243610186929099236113, 5.62259116363956020376455663683, 6.25525089783940231261702925989, 7.59549932625553931031475947377, 8.283352481452576656131694320219, 8.922037140997386799693330178878, 9.633505848472215432000699541525, 11.31988061405075497715663962488
