| L(s) = 1 | + (0.557 + 1.29i)2-s + (1.75 + 1.94i)3-s + (−1.37 + 1.44i)4-s + (−1.65 − 2.87i)5-s + (−1.55 + 3.36i)6-s + (3.31 − 0.348i)7-s + (−2.65 − 0.984i)8-s + (−0.405 + 3.86i)9-s + (2.80 − 3.75i)10-s + (−3.87 − 1.72i)11-s + (−5.24 − 0.144i)12-s + (−0.203 − 0.957i)13-s + (2.30 + 4.11i)14-s + (2.68 − 8.27i)15-s + (−0.197 − 3.99i)16-s + (−0.477 − 1.07i)17-s + ⋯ |
| L(s) = 1 | + (0.394 + 0.919i)2-s + (1.01 + 1.12i)3-s + (−0.689 + 0.724i)4-s + (−0.741 − 1.28i)5-s + (−0.635 + 1.37i)6-s + (1.25 − 0.131i)7-s + (−0.937 − 0.348i)8-s + (−0.135 + 1.28i)9-s + (0.887 − 1.18i)10-s + (−1.16 − 0.520i)11-s + (−1.51 − 0.0418i)12-s + (−0.0564 − 0.265i)13-s + (0.614 + 1.09i)14-s + (0.693 − 2.13i)15-s + (−0.0494 − 0.998i)16-s + (−0.115 − 0.260i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 124 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0675 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.00970 + 1.08041i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.00970 + 1.08041i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.557 - 1.29i)T \) |
| 31 | \( 1 + (-2.31 - 5.06i)T \) |
| good | 3 | \( 1 + (-1.75 - 1.94i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + (1.65 + 2.87i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-3.31 + 0.348i)T + (6.84 - 1.45i)T^{2} \) |
| 11 | \( 1 + (3.87 + 1.72i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.203 + 0.957i)T + (-11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (0.477 + 1.07i)T + (-11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.801 - 3.77i)T + (-17.3 - 7.72i)T^{2} \) |
| 23 | \( 1 + (-3.48 - 2.53i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.15 + 0.700i)T + (23.4 - 17.0i)T^{2} \) |
| 37 | \( 1 + (8.35 + 4.82i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.09 - 3.43i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (-1.10 - 0.235i)T + (39.2 + 17.4i)T^{2} \) |
| 47 | \( 1 + (10.9 + 3.54i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-8.89 - 0.934i)T + (51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (4.85 - 4.37i)T + (6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + 9.56iT - 61T^{2} \) |
| 67 | \( 1 + (3.60 - 2.07i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.02 + 0.317i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (3.77 - 8.48i)T + (-48.8 - 54.2i)T^{2} \) |
| 79 | \( 1 + (-6.70 + 2.98i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-6.18 + 6.86i)T + (-8.67 - 82.5i)T^{2} \) |
| 89 | \( 1 + (-0.174 - 0.240i)T + (-27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-11.7 + 8.55i)T + (29.9 - 92.2i)T^{2} \) |
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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
−13.95916172011702974957382849190, −12.99290348348381416606914836003, −11.80472789672531293446689580514, −10.35733122307475392595580557450, −8.903009031512960632067625577413, −8.345388746362311092088085364496, −7.71348496170943798990400845508, −5.19855055778913900824571239372, −4.69482980634673880809170232117, −3.44843274254519975110343042028,
2.09803639239358848044323172551, 3.05344651463515513362648432421, 4.76746065209801325402788201148, 6.79984022704067404840456929719, 7.81229604869769080790987867285, 8.676779457634164858190261027695, 10.38814450860669857143048712345, 11.22285257817012228162839557111, 12.13373675615545497051652433243, 13.22553543076278549804591357695
