L(s) = 1 | + (−1.81 − 1.04i)5-s + (0.143 − 0.0829i)7-s + (0.784 + 1.35i)11-s + (−1.93 + 3.34i)13-s − 5.27i·17-s − 8.05i·19-s + (2.67 − 4.63i)23-s + (−0.298 − 0.516i)25-s + (−6.75 + 3.89i)29-s + (−2.10 − 1.21i)31-s − 0.348·35-s − 8.53·37-s + (−2.47 − 1.43i)41-s + (−3.42 + 1.97i)43-s + (−3.68 − 6.38i)47-s + ⋯ |
L(s) = 1 | + (−0.812 − 0.469i)5-s + (0.0543 − 0.0313i)7-s + (0.236 + 0.409i)11-s + (−0.535 + 0.928i)13-s − 1.27i·17-s − 1.84i·19-s + (0.557 − 0.966i)23-s + (−0.0596 − 0.103i)25-s + (−1.25 + 0.724i)29-s + (−0.378 − 0.218i)31-s − 0.0588·35-s − 1.40·37-s + (−0.387 − 0.223i)41-s + (−0.521 + 0.301i)43-s + (−0.537 − 0.931i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.626 + 0.779i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.305725 - 0.637787i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.305725 - 0.637787i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.81 + 1.04i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.143 + 0.0829i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.784 - 1.35i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (1.93 - 3.34i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.27iT - 17T^{2} \) |
| 19 | \( 1 + 8.05iT - 19T^{2} \) |
| 23 | \( 1 + (-2.67 + 4.63i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (6.75 - 3.89i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2.10 + 1.21i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8.53T + 37T^{2} \) |
| 41 | \( 1 + (2.47 + 1.43i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.42 - 1.97i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (3.68 + 6.38i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 2.40iT - 53T^{2} \) |
| 59 | \( 1 + (-5.49 + 9.52i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (7.11 + 12.3i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.45 - 0.841i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.5T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + (6.31 - 3.64i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.35 - 2.34i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 2.40iT - 89T^{2} \) |
| 97 | \( 1 + (-0.903 - 1.56i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.579652401827903697580959025483, −9.127568815532556942568703978733, −8.225481384964708098073124639115, −7.08891855781848595954420779032, −6.81261407858670863181508407757, −5.02445635975803717269161046150, −4.68853425113025401774430832241, −3.45795223066490395799017081288, −2.13684585800862854127235328180, −0.33599748524413812670043274571,
1.69991860332330337703728626190, 3.39229887158456488939316601501, 3.80230063815657345834230201258, 5.31658488315028031537451213497, 6.05221757202415264345471683709, 7.25119768481346904623849459374, 7.88649552924365948900770828750, 8.584514576351717964006233774469, 9.791668936065031369641329378648, 10.47509087009774374278153840110