| L(s) = 1 | + (0.00745 − 0.00508i)2-s + (−2.32 + 2.15i)3-s + (−0.730 + 1.86i)4-s + (2.27 + 0.701i)5-s + (−0.00635 + 0.0278i)6-s + (−1.04 − 2.42i)7-s + (0.00802 + 0.0351i)8-s + (0.524 − 7.00i)9-s + (0.0205 − 0.00632i)10-s + (−0.443 − 5.91i)11-s + (−2.31 − 5.89i)12-s + (−0.900 − 0.433i)13-s + (−0.0201 − 0.0127i)14-s + (−6.78 + 3.26i)15-s + (−2.93 − 2.72i)16-s + (3.33 + 0.502i)17-s + ⋯ |
| L(s) = 1 | + (0.00526 − 0.00359i)2-s + (−1.33 + 1.24i)3-s + (−0.365 + 0.930i)4-s + (1.01 + 0.313i)5-s + (−0.00259 + 0.0113i)6-s + (−0.396 − 0.918i)7-s + (0.00283 + 0.0124i)8-s + (0.174 − 2.33i)9-s + (0.00648 − 0.00199i)10-s + (−0.133 − 1.78i)11-s + (−0.667 − 1.70i)12-s + (−0.249 − 0.120i)13-s + (−0.00538 − 0.00341i)14-s + (−1.75 + 0.843i)15-s + (−0.732 − 0.680i)16-s + (0.807 + 0.121i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.815 + 0.578i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.585068 - 0.186598i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.585068 - 0.186598i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (1.04 + 2.42i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| good | 2 | \( 1 + (-0.00745 + 0.00508i)T + (0.730 - 1.86i)T^{2} \) |
| 3 | \( 1 + (2.32 - 2.15i)T + (0.224 - 2.99i)T^{2} \) |
| 5 | \( 1 + (-2.27 - 0.701i)T + (4.13 + 2.81i)T^{2} \) |
| 11 | \( 1 + (0.443 + 5.91i)T + (-10.8 + 1.63i)T^{2} \) |
| 17 | \( 1 + (-3.33 - 0.502i)T + (16.2 + 5.01i)T^{2} \) |
| 19 | \( 1 + (-1.92 + 3.33i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.91 - 0.288i)T + (21.9 - 6.77i)T^{2} \) |
| 29 | \( 1 + (3.47 - 4.35i)T + (-6.45 - 28.2i)T^{2} \) |
| 31 | \( 1 + (3.31 + 5.73i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (0.404 + 1.03i)T + (-27.1 + 25.1i)T^{2} \) |
| 41 | \( 1 + (1.30 + 5.73i)T + (-36.9 + 17.7i)T^{2} \) |
| 43 | \( 1 + (1.13 - 4.98i)T + (-38.7 - 18.6i)T^{2} \) |
| 47 | \( 1 + (-0.460 + 0.313i)T + (17.1 - 43.7i)T^{2} \) |
| 53 | \( 1 + (-1.57 + 4.00i)T + (-38.8 - 36.0i)T^{2} \) |
| 59 | \( 1 + (7.46 - 2.30i)T + (48.7 - 33.2i)T^{2} \) |
| 61 | \( 1 + (4.20 + 10.7i)T + (-44.7 + 41.4i)T^{2} \) |
| 67 | \( 1 + (7.65 + 13.2i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-4.91 - 6.16i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-10.6 - 7.28i)T + (26.6 + 67.9i)T^{2} \) |
| 79 | \( 1 + (0.935 - 1.62i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-9.10 + 4.38i)T + (51.7 - 64.8i)T^{2} \) |
| 89 | \( 1 + (0.516 - 6.88i)T + (-88.0 - 13.2i)T^{2} \) |
| 97 | \( 1 - 4.27T + 97T^{2} \) |
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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
−10.64744038176458044919250523504, −9.660424846122318635400428243097, −9.209934797504251952482186090216, −7.84256614219141944057138046675, −6.63674196594126324029905239365, −5.81386502108082348864201014057, −5.07228861748420031903168768823, −3.83360699148792988259306728044, −3.21314645936109225530460859534, −0.40673421911219556492391482437,
1.49189328439613313905316976024, 2.10568117583976754678175585277, 4.77740454629807504937629410187, 5.50791950397653070673246447672, 5.94984452830649229474161832420, 6.83486379492887495606945913468, 7.76196656671087496914632647219, 9.244285522935846840957846989586, 9.903144914298603399441456549128, 10.50707814674719408845030530895
