L(s) = 1 | + (1.5 + 2.59i)3-s + (1 + 1.73i)4-s + (0.5 − 0.866i)5-s + (−3 + 5.19i)9-s + (0.5 + 0.866i)11-s + (−3 + 5.19i)12-s − 4·13-s + 3·15-s + (−1.99 + 3.46i)16-s + (−1 − 1.73i)17-s + (3 − 5.19i)19-s + 1.99·20-s + (2.5 − 4.33i)23-s + (2 + 3.46i)25-s − 9·27-s + ⋯ |
L(s) = 1 | + (0.866 + 1.49i)3-s + (0.5 + 0.866i)4-s + (0.223 − 0.387i)5-s + (−1 + 1.73i)9-s + (0.150 + 0.261i)11-s + (−0.866 + 1.49i)12-s − 1.10·13-s + 0.774·15-s + (−0.499 + 0.866i)16-s + (−0.242 − 0.420i)17-s + (0.688 − 1.19i)19-s + 0.447·20-s + (0.521 − 0.902i)23-s + (0.400 + 0.692i)25-s − 1.73·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16016 + 1.74413i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16016 + 1.74413i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.5 - 2.59i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 4T + 13T^{2} \) |
| 17 | \( 1 + (1 + 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3 + 5.19i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.5 + 4.33i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 10T + 29T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.5 + 4.33i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 2T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-1 + 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.5 - 2.59i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 + (5 + 8.66i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3 - 5.19i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-7.5 + 12.9i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 5T + 97T^{2} \) |
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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
−10.96379487214046019153893921882, −10.06986691887802758286254371549, −9.234550385089923451083999202315, −8.711758289428788578337926299125, −7.70013696910997906183928782221, −6.74715643226183245994784074643, −4.95940787832346965196735567374, −4.52780090684663945905550295322, −3.16751200351392269646216578513, −2.52435428238875203424119709899,
1.20444238805381176355335625363, 2.25429689320404033145529027823, 3.19063344524637021861846921416, 5.16156503470598894836084540918, 6.36229882575383413690779574767, 6.82264597078474354697976335360, 7.74027183513698377277798092267, 8.575515160602076436545018895227, 9.737992586156509145277822006447, 10.38002010556796977932446041898