L(s) = 1 | + (0.110 + 0.138i)2-s + (−0.222 − 0.974i)3-s + (0.438 − 1.91i)4-s + (−2.15 − 2.69i)5-s + (0.110 − 0.138i)6-s + (0.844 + 3.70i)7-s + (0.633 − 0.304i)8-s + (−0.900 + 0.433i)9-s + (0.135 − 0.595i)10-s + (3.84 + 1.85i)11-s − 1.96·12-s + (4.18 + 2.01i)13-s + (−0.419 + 0.525i)14-s + (−2.15 + 2.69i)15-s + (−3.43 − 1.65i)16-s − 3.07·17-s + ⋯ |
L(s) = 1 | + (0.0780 + 0.0978i)2-s + (−0.128 − 0.562i)3-s + (0.219 − 0.959i)4-s + (−0.962 − 1.20i)5-s + (0.0450 − 0.0565i)6-s + (0.319 + 1.39i)7-s + (0.223 − 0.107i)8-s + (−0.300 + 0.144i)9-s + (0.0430 − 0.188i)10-s + (1.15 + 0.558i)11-s − 0.568·12-s + (1.16 + 0.558i)13-s + (−0.111 + 0.140i)14-s + (−0.555 + 0.696i)15-s + (−0.858 − 0.413i)16-s − 0.745·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.474 + 0.880i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.813378 - 0.485387i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.813378 - 0.485387i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.222 + 0.974i)T \) |
| 29 | \( 1 + (1.41 - 5.19i)T \) |
good | 2 | \( 1 + (-0.110 - 0.138i)T + (-0.445 + 1.94i)T^{2} \) |
| 5 | \( 1 + (2.15 + 2.69i)T + (-1.11 + 4.87i)T^{2} \) |
| 7 | \( 1 + (-0.844 - 3.70i)T + (-6.30 + 3.03i)T^{2} \) |
| 11 | \( 1 + (-3.84 - 1.85i)T + (6.85 + 8.60i)T^{2} \) |
| 13 | \( 1 + (-4.18 - 2.01i)T + (8.10 + 10.1i)T^{2} \) |
| 17 | \( 1 + 3.07T + 17T^{2} \) |
| 19 | \( 1 + (-0.799 + 3.50i)T + (-17.1 - 8.24i)T^{2} \) |
| 23 | \( 1 + (0.270 - 0.339i)T + (-5.11 - 22.4i)T^{2} \) |
| 31 | \( 1 + (2.81 + 3.53i)T + (-6.89 + 30.2i)T^{2} \) |
| 37 | \( 1 + (5.70 - 2.74i)T + (23.0 - 28.9i)T^{2} \) |
| 41 | \( 1 + 1.97T + 41T^{2} \) |
| 43 | \( 1 + (0.156 - 0.196i)T + (-9.56 - 41.9i)T^{2} \) |
| 47 | \( 1 + (-4.33 - 2.08i)T + (29.3 + 36.7i)T^{2} \) |
| 53 | \( 1 + (-6.83 - 8.57i)T + (-11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + 6.06T + 59T^{2} \) |
| 61 | \( 1 + (0.843 + 3.69i)T + (-54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 + (-4.74 + 2.28i)T + (41.7 - 52.3i)T^{2} \) |
| 71 | \( 1 + (-5.25 - 2.53i)T + (44.2 + 55.5i)T^{2} \) |
| 73 | \( 1 + (6.57 - 8.24i)T + (-16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 + (9.04 - 4.35i)T + (49.2 - 61.7i)T^{2} \) |
| 83 | \( 1 + (-1.57 + 6.92i)T + (-74.7 - 36.0i)T^{2} \) |
| 89 | \( 1 + (8.71 + 10.9i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + (-3.18 + 13.9i)T + (-87.3 - 42.0i)T^{2} \) |
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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
−14.05837496491112017887779645986, −12.78559436684390851059278183530, −11.80338145237446541604635134830, −11.25771287304732653181834233074, −9.115843622506061739692580368098, −8.704512754058978002650269533933, −6.95744927152153872264786266546, −5.70557473956735926140824629900, −4.48230233580441468873033632859, −1.59293336240047390911440003872,
3.60024090728967690911123052478, 3.91975088207769563914483179072, 6.53872968387328789410731196416, 7.51929970288157838660452618717, 8.587450755224876100517384035861, 10.51994037809613692176538014627, 11.10553710782365320729925803066, 11.86858572639326942896405778249, 13.49423844387374654129049164042, 14.34523316707992453157793477283