| L(s) = 1 | + (−0.399 + 1.74i)2-s + (0.900 − 0.433i)3-s + (−1.10 − 0.529i)4-s + (−0.299 + 1.31i)5-s + (0.399 + 1.74i)6-s + (1.00 − 0.483i)7-s + (−0.871 + 1.09i)8-s + (0.623 − 0.781i)9-s + (−2.17 − 1.04i)10-s + (−1.53 − 1.93i)11-s − 1.22·12-s + (−1.75 − 2.20i)13-s + (0.444 + 1.94i)14-s + (0.299 + 1.31i)15-s + (−3.08 − 3.87i)16-s + 5.50·17-s + ⋯ |
| L(s) = 1 | + (−0.282 + 1.23i)2-s + (0.520 − 0.250i)3-s + (−0.550 − 0.264i)4-s + (−0.133 + 0.586i)5-s + (0.163 + 0.714i)6-s + (0.379 − 0.182i)7-s + (−0.308 + 0.386i)8-s + (0.207 − 0.260i)9-s + (−0.687 − 0.331i)10-s + (−0.464 − 0.582i)11-s − 0.352·12-s + (−0.487 − 0.610i)13-s + (0.118 + 0.520i)14-s + (0.0772 + 0.338i)15-s + (−0.771 − 0.967i)16-s + 1.33·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0372 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.709558 + 0.683571i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.709558 + 0.683571i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.900 + 0.433i)T \) |
| 29 | \( 1 + (5.29 - 0.981i)T \) |
| good | 2 | \( 1 + (0.399 - 1.74i)T + (-1.80 - 0.867i)T^{2} \) |
| 5 | \( 1 + (0.299 - 1.31i)T + (-4.50 - 2.16i)T^{2} \) |
| 7 | \( 1 + (-1.00 + 0.483i)T + (4.36 - 5.47i)T^{2} \) |
| 11 | \( 1 + (1.53 + 1.93i)T + (-2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (1.75 + 2.20i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 - 5.50T + 17T^{2} \) |
| 19 | \( 1 + (-0.318 - 0.153i)T + (11.8 + 14.8i)T^{2} \) |
| 23 | \( 1 + (1.18 + 5.18i)T + (-20.7 + 9.97i)T^{2} \) |
| 31 | \( 1 + (-0.990 + 4.33i)T + (-27.9 - 13.4i)T^{2} \) |
| 37 | \( 1 + (6.00 - 7.52i)T + (-8.23 - 36.0i)T^{2} \) |
| 41 | \( 1 + 9.05T + 41T^{2} \) |
| 43 | \( 1 + (-2.01 - 8.82i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (-2.24 - 2.81i)T + (-10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (-1.46 + 6.41i)T + (-47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + 6.26T + 59T^{2} \) |
| 61 | \( 1 + (0.106 - 0.0512i)T + (38.0 - 47.6i)T^{2} \) |
| 67 | \( 1 + (-8.91 + 11.1i)T + (-14.9 - 65.3i)T^{2} \) |
| 71 | \( 1 + (-4.68 - 5.86i)T + (-15.7 + 69.2i)T^{2} \) |
| 73 | \( 1 + (-3.54 - 15.5i)T + (-65.7 + 31.6i)T^{2} \) |
| 79 | \( 1 + (-0.160 + 0.200i)T + (-17.5 - 77.0i)T^{2} \) |
| 83 | \( 1 + (-5.52 - 2.65i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (1.66 - 7.31i)T + (-80.1 - 38.6i)T^{2} \) |
| 97 | \( 1 + (6.37 + 3.07i)T + (60.4 + 75.8i)T^{2} \) |
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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
−14.66609146321388820445480692163, −13.89833404375414612127693639093, −12.49862358696200197818924384935, −11.14166482605076096030060578330, −9.833408095384208736425607964526, −8.314037887750172186842692903276, −7.73673413998218321588103746075, −6.63102492328295062507112919443, −5.28678870161059551461664721660, −3.02280751128944763674251069583,
1.91113329301265944928779355277, 3.59486991487819349566625928273, 5.15520813837550531855458318983, 7.37124917646129727597585952061, 8.749281084830674980893654728294, 9.678802075328873372790333170407, 10.57970530246585255993190735620, 11.91049142222315017987804321054, 12.47299811079381710247712652747, 13.74349436034626932759294383639
