L(s) = 1 | + (3.86 − 2.23i)3-s + (7.33 + 4.23i)5-s + (5.44 − 9.43i)9-s + (1.94 + 3.37i)11-s + 19.1i·13-s + 37.7·15-s + (−11.5 + 6.69i)17-s + (5.80 + 3.35i)19-s + (13 − 22.5i)23-s + (23.3 + 40.5i)25-s − 8.47i·27-s − 11.7·29-s + (31.6 − 18.2i)31-s + (15.0 + 8.69i)33-s + (16 − 27.7i)37-s + ⋯ |
L(s) = 1 | + (1.28 − 0.743i)3-s + (1.46 + 0.847i)5-s + (0.605 − 1.04i)9-s + (0.177 + 0.307i)11-s + 1.47i·13-s + 2.51·15-s + (−0.681 + 0.393i)17-s + (0.305 + 0.176i)19-s + (0.565 − 0.978i)23-s + (0.935 + 1.62i)25-s − 0.313i·27-s − 0.406·29-s + (1.02 − 0.590i)31-s + (0.456 + 0.263i)33-s + (0.432 − 0.748i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.990 - 0.134i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(3.812779095\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.812779095\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-3.86 + 2.23i)T + (4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (-7.33 - 4.23i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-1.94 - 3.37i)T + (-60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 19.1iT - 169T^{2} \) |
| 17 | \( 1 + (11.5 - 6.69i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-5.80 - 3.35i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (-13 + 22.5i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 11.7T + 841T^{2} \) |
| 31 | \( 1 + (-31.6 + 18.2i)T + (480.5 - 832. i)T^{2} \) |
| 37 | \( 1 + (-16 + 27.7i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 20.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 79.2T + 1.84e3T^{2} \) |
| 47 | \( 1 + (12.3 + 7.15i)T + (1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-6.89 - 11.9i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (7.31 - 4.22i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (27.4 + 15.8i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (15.6 + 27.1i)T + (-2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + 95.5T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-15.8 + 9.15i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-39.8 + 69.1i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 141. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (-100. - 58.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 137. iT - 9.40e3T^{2} \) |
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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
−9.877812906987219725265186643246, −9.195445018652352965581866951926, −8.591039862672084658477687658023, −7.37820102756633538526466018183, −6.71096963947592303631575575807, −6.10035068276798072945181896012, −4.56971675846303440709608297340, −3.21808896132045697389046818007, −2.21363724711922804010423105324, −1.72744567005346606166057153705,
1.23877353591711126642093100427, 2.56486258068134860310039869370, 3.34863674684766738874998806124, 4.75193860900905846172644387553, 5.38713683427236824975781013831, 6.47815976691327143569098402555, 7.889009573714961033713267024866, 8.603612882429504750590348086180, 9.260839248052676202952098092465, 9.857136534587604559185851426064