L(s) = 1 | + (−1.97 + 1.14i)2-s + (−2.17 + 1.25i)3-s + (1.61 − 2.79i)4-s + (2.86 − 4.96i)6-s − 3.50i·7-s + 2.79i·8-s + (1.64 − 2.84i)9-s − 4.50·11-s + 8.07i·12-s + (4.33 + 2.5i)13-s + (4.00 + 6.94i)14-s + (0.0316 + 0.0547i)16-s + (−0.137 + 0.0793i)17-s + 7.50i·18-s + (4.26 + 0.920i)19-s + ⋯ |
L(s) = 1 | + (−1.39 + 0.807i)2-s + (−1.25 + 0.723i)3-s + (0.805 − 1.39i)4-s + (1.16 − 2.02i)6-s − 1.32i·7-s + 0.987i·8-s + (0.547 − 0.948i)9-s − 1.35·11-s + 2.33i·12-s + (1.20 + 0.693i)13-s + (1.07 + 1.85i)14-s + (0.00790 + 0.0136i)16-s + (−0.0333 + 0.0192i)17-s + 1.76i·18-s + (0.977 + 0.211i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.884 - 0.466i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0650988 + 0.262795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0650988 + 0.262795i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 + (-4.26 - 0.920i)T \) |
good | 2 | \( 1 + (1.97 - 1.14i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (2.17 - 1.25i)T + (1.5 - 2.59i)T^{2} \) |
| 7 | \( 1 + 3.50iT - 7T^{2} \) |
| 11 | \( 1 + 4.50T + 11T^{2} \) |
| 13 | \( 1 + (-4.33 - 2.5i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.137 - 0.0793i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (1.00 + 0.579i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (1.75 - 3.03i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 - 10.9iT - 37T^{2} \) |
| 41 | \( 1 + (3.03 + 5.26i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.89 - 1.67i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.65 - 1.53i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.97 - 2.87i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.53 - 2.65i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.436 + 0.756i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.31 + 4.22i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-8.11 - 14.0i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (6.19 - 3.57i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (5.06 + 8.76i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.85iT - 83T^{2} \) |
| 89 | \( 1 + (0.556 - 0.963i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.40 - 0.809i)T + (48.5 - 84.0i)T^{2} \) |
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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
−10.87280672682545611044340277150, −10.39187353277291650432306637159, −9.836408707394721748854110969710, −8.677106249654338087762374588911, −7.70375836801689492122485772897, −6.90965040752628556359587439651, −5.99963363457053245665093239141, −5.05139435061930171573947220859, −3.80741101720927040588792293694, −1.08429278374037635509062596045,
0.37708249288552300273319581125, 1.85252958895028047670747355571, 3.04078399755626853139889848622, 5.41514099913384297507747464977, 5.79759106603790936472737967869, 7.23717958361812601533207317094, 8.076908395850461028244962850638, 8.868537697808641011754112803265, 9.890149042148019190927219641859, 10.83544308951075331680898620611