| 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
−10.50707814674719408845030530895, −9.903144914298603399441456549128, −9.244285522935846840957846989586, −7.76196656671087496914632647219, −6.83486379492887495606945913468, −5.94984452830649229474161832420, −5.50791950397653070673246447672, −4.77740454629807504937629410187, −2.10568117583976754678175585277, −1.49189328439613313905316976024,
0.40673421911219556492391482437, 3.21314645936109225530460859534, 3.83360699148792988259306728044, 5.07228861748420031903168768823, 5.81386502108082348864201014057, 6.63674196594126324029905239365, 7.84256614219141944057138046675, 9.209934797504251952482186090216, 9.660424846122318635400428243097, 10.64744038176458044919250523504
