| L(s) = 1 | + (1.99 − 0.0127i)2-s + (−1.22 + 4.55i)3-s + (3.99 − 0.0508i)4-s + (3.64 + 3.42i)5-s + (−2.38 + 9.13i)6-s + (−5.07 − 4.82i)7-s + (7.99 − 0.152i)8-s + (−11.4 − 6.63i)9-s + (7.33 + 6.79i)10-s + (−9.58 + 5.53i)11-s + (−4.65 + 18.2i)12-s + (8.35 − 8.35i)13-s + (−10.2 − 9.58i)14-s + (−20.0 + 12.4i)15-s + (15.9 − 0.406i)16-s + (0.550 − 2.05i)17-s + ⋯ |
| L(s) = 1 | + (0.999 − 0.00635i)2-s + (−0.407 + 1.51i)3-s + (0.999 − 0.0127i)4-s + (0.728 + 0.684i)5-s + (−0.397 + 1.52i)6-s + (−0.724 − 0.688i)7-s + (0.999 − 0.0190i)8-s + (−1.27 − 0.737i)9-s + (0.733 + 0.679i)10-s + (−0.871 + 0.503i)11-s + (−0.387 + 1.52i)12-s + (0.642 − 0.642i)13-s + (−0.729 − 0.684i)14-s + (−1.33 + 0.828i)15-s + (0.999 − 0.0254i)16-s + (0.0324 − 0.120i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.0871 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73681 + 1.59146i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73681 + 1.59146i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 + 0.0127i)T \) |
| 5 | \( 1 + (-3.64 - 3.42i)T \) |
| 7 | \( 1 + (5.07 + 4.82i)T \) |
| good | 3 | \( 1 + (1.22 - 4.55i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (9.58 - 5.53i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-8.35 + 8.35i)T - 169iT^{2} \) |
| 17 | \( 1 + (-0.550 + 2.05i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (-27.9 - 16.1i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (8.01 + 29.9i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 + 23.6iT - 841T^{2} \) |
| 31 | \( 1 + (-3.58 - 6.20i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (10.1 + 37.6i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 - 2.34iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-27.8 + 27.8i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (11.5 + 42.9i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (19.6 - 73.1i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (51.4 - 29.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-3.99 - 2.30i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-7.65 + 28.5i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 8.97iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (75.2 + 20.1i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (-48.6 + 84.2i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (15.0 + 15.0i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (41.5 - 71.9i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-53.2 - 53.2i)T + 9.40e3iT^{2} \) |
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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
−13.37018980882389923416261654887, −12.17016339292999133743151746479, −10.69352435345233420720888553708, −10.47063745386860241086417143061, −9.615255675802163656350704047784, −7.52311107974709118382220992975, −6.13745430160910948032048777585, −5.31628909319279406481430976425, −3.99262212755976793888027779630, −2.94363843921494763503322220580,
1.46703371651842561477652786399, 2.91972531586012718452287917023, 5.26699250546667642484497115946, 5.95972156484587885415260062660, 6.89577370119298024179283630479, 8.123875521808300009768091911481, 9.566791078131886762206240835081, 11.23504584118231658661680514945, 12.01331692844023032006105234669, 12.91770807277969681524192461110
