L(s) = 1 | + (0.0819 + 1.99i)2-s + (4.93 + 2.84i)3-s + (−3.98 + 0.327i)4-s + (1.11 + 1.93i)5-s + (−5.28 + 10.0i)6-s + (−6.48 − 2.62i)7-s + (−0.981 − 7.93i)8-s + (11.7 + 20.3i)9-s + (−3.77 + 2.39i)10-s + (−5.26 − 3.03i)11-s + (−20.5 − 9.73i)12-s + 11.5·13-s + (4.71 − 13.1i)14-s + 12.7i·15-s + (15.7 − 2.61i)16-s + (2.84 − 4.92i)17-s + ⋯ |
L(s) = 1 | + (0.0409 + 0.999i)2-s + (1.64 + 0.949i)3-s + (−0.996 + 0.0818i)4-s + (0.223 + 0.387i)5-s + (−0.881 + 1.68i)6-s + (−0.926 − 0.375i)7-s + (−0.122 − 0.992i)8-s + (1.30 + 2.25i)9-s + (−0.377 + 0.239i)10-s + (−0.478 − 0.276i)11-s + (−1.71 − 0.811i)12-s + 0.888·13-s + (0.337 − 0.941i)14-s + 0.849i·15-s + (0.986 − 0.163i)16-s + (0.167 − 0.289i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.897643 + 1.88255i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.897643 + 1.88255i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.0819 - 1.99i)T \) |
| 5 | \( 1 + (-1.11 - 1.93i)T \) |
| 7 | \( 1 + (6.48 + 2.62i)T \) |
good | 3 | \( 1 + (-4.93 - 2.84i)T + (4.5 + 7.79i)T^{2} \) |
| 11 | \( 1 + (5.26 + 3.03i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 - 11.5T + 169T^{2} \) |
| 17 | \( 1 + (-2.84 + 4.92i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-29.1 + 16.8i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (17.6 - 10.1i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 10.7T + 841T^{2} \) |
| 31 | \( 1 + (14.2 + 8.24i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (20.3 + 35.2i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 - 41.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.10iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-12.6 + 7.30i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-15.0 + 26.1i)T + (-1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + (29.1 + 16.8i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (40.4 + 70.1i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (7.33 + 4.23i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 21.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (13.0 - 22.5i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-7.65 + 4.42i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 102. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + (26.6 + 46.0i)T + (-3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 176.T + 9.40e3T^{2} \) |
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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
−13.77249957172517759998786087380, −13.00249283066076497360764003982, −10.75052099940064465140511917051, −9.646138882253050167237917979091, −9.228401358212848625040369099011, −7.997220732737059517191197137937, −7.14036191459358034447705618713, −5.50896575355521718438958246310, −3.93964436479243757557425690678, −3.07978266794064811234431069609,
1.44614630433437595063060607443, 2.80979869255570838598899776115, 3.78423266856241104843699088339, 5.92750100466910493539498483003, 7.59762012540619108760863100607, 8.577398305602779244573452122710, 9.360451906618751924421663442315, 10.18140421733931837750779719720, 12.07305647355350201827602388260, 12.63394758051421365619300967265