L(s) = 1 | + (−1.21 + 1.22i)3-s + (−1.40 + 2.43i)5-s + (−2.08 − 1.62i)7-s + (−0.0238 − 2.99i)9-s + (−4.74 + 2.74i)11-s + 1.35i·13-s + (−1.27 − 4.69i)15-s + (2.88 + 5.00i)17-s + (1.71 + 0.992i)19-s + (4.54 − 0.579i)21-s + (2.09 + 1.21i)23-s + (−1.44 − 2.49i)25-s + (3.71 + 3.63i)27-s − 7.05i·29-s + (−3.07 + 1.77i)31-s + ⋯ |
L(s) = 1 | + (−0.704 + 0.709i)3-s + (−0.627 + 1.08i)5-s + (−0.788 − 0.615i)7-s + (−0.00795 − 0.999i)9-s + (−1.43 + 0.826i)11-s + 0.376i·13-s + (−0.329 − 1.21i)15-s + (0.700 + 1.21i)17-s + (0.394 + 0.227i)19-s + (0.991 − 0.126i)21-s + (0.437 + 0.252i)23-s + (−0.288 − 0.499i)25-s + (0.715 + 0.698i)27-s − 1.31i·29-s + (−0.552 + 0.318i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 168 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.128791 + 0.480776i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.128791 + 0.480776i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.21 - 1.22i)T \) |
| 7 | \( 1 + (2.08 + 1.62i)T \) |
good | 5 | \( 1 + (1.40 - 2.43i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (4.74 - 2.74i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.35iT - 13T^{2} \) |
| 17 | \( 1 + (-2.88 - 5.00i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.71 - 0.992i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.09 - 1.21i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.05iT - 29T^{2} \) |
| 31 | \( 1 + (3.07 - 1.77i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.14 - 3.71i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 1.81T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 + (0.201 - 0.348i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.28 - 3.04i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.28 - 2.22i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.75 + 2.74i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.45 - 5.97i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.08iT - 71T^{2} \) |
| 73 | \( 1 + (0.295 - 0.170i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 2.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 + (0.576 - 0.998i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−13.02189075105338747768138649530, −12.12356357336383194579016336356, −10.93138462953627401086163953151, −10.39190421281249939339701453135, −9.615870259833836478215065935794, −7.79654837456688486563625304802, −6.91249305199321856535848625322, −5.74174983219573723457030630915, −4.23907614243485503300542091550, −3.15169494529441115946275292868,
0.49503094609902077882147536377, 2.94124069348704167821615651012, 5.06697063244296710843847252222, 5.65913838288105539841790727484, 7.22133174025367883757898676071, 8.147231868546498121494030643569, 9.182414199763645268414004668611, 10.59712744731384832730724131405, 11.59532750932295647688508743935, 12.62289298758542205272319297801