L(s) = 1 | + (−1.31 + 2.27i)2-s + 3-s + (−2.44 − 4.23i)4-s + (0.734 + 1.27i)5-s + (−1.31 + 2.27i)6-s + (2.40 + 1.09i)7-s + 7.59·8-s + 9-s − 3.85·10-s + 4.48·11-s + (−2.44 − 4.23i)12-s + (−3.58 − 0.374i)13-s + (−5.65 + 4.03i)14-s + (0.734 + 1.27i)15-s + (−5.08 + 8.80i)16-s + (1.81 + 3.14i)17-s + ⋯ |
L(s) = 1 | + (−0.928 + 1.60i)2-s + 0.577·3-s + (−1.22 − 2.11i)4-s + (0.328 + 0.569i)5-s + (−0.535 + 0.928i)6-s + (0.910 + 0.414i)7-s + 2.68·8-s + 0.333·9-s − 1.21·10-s + 1.35·11-s + (−0.706 − 1.22i)12-s + (−0.994 − 0.103i)13-s + (−1.51 + 1.07i)14-s + (0.189 + 0.328i)15-s + (−1.27 + 2.20i)16-s + (0.440 + 0.762i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.431331 + 0.952975i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.431331 + 0.952975i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 7 | \( 1 + (-2.40 - 1.09i)T \) |
| 13 | \( 1 + (3.58 + 0.374i)T \) |
good | 2 | \( 1 + (1.31 - 2.27i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.734 - 1.27i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 17 | \( 1 + (-1.81 - 3.14i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 + (3.87 - 6.71i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.98 + 6.90i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (0.552 - 0.957i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.57 + 2.72i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.11 - 1.92i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-2.54 + 4.41i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.69 - 2.94i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (4.25 - 7.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (7.22 + 12.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 7.62T + 61T^{2} \) |
| 67 | \( 1 - 4.67T + 67T^{2} \) |
| 71 | \( 1 + (-1.24 + 2.15i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.03 + 5.24i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.59 + 6.22i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.30T + 83T^{2} \) |
| 89 | \( 1 + (3.18 - 5.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.02 - 6.96i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.25822249475599349359559766642, −10.93466044570791947453319683065, −9.788130341060995088150145377700, −9.250700631372466240807798751156, −8.177223049719286695411838408453, −7.57111193813465832071555234211, −6.49019414554053376084814797548, −5.63492089424126354326352686698, −4.23902982354403949997243311507, −1.83989964732314826392000743735,
1.22229014842293744782410772822, 2.33943966874041079875582125517, 3.85962491786833665956446709060, 4.79847050372398418286204034097, 7.11836519443937516678284389425, 8.198381684453272470334225321466, 8.978964478867718458537208373022, 9.620525352049623475401715794013, 10.54516203879631099721309059000, 11.50334303458542488245821225282