L(s) = 1 | + (−0.397 + 1.35i)2-s + (0.556 − 0.963i)3-s + (−1.68 − 1.07i)4-s + (0.866 − 0.5i)5-s + (1.08 + 1.13i)6-s + (2.32 + 1.26i)7-s + (2.13 − 1.85i)8-s + (0.880 + 1.52i)9-s + (0.334 + 1.37i)10-s + (1.48 + 0.856i)11-s + (−1.97 + 1.02i)12-s − 2.45i·13-s + (−2.63 + 2.65i)14-s − 1.11i·15-s + (1.67 + 3.63i)16-s + (−5.38 − 3.10i)17-s + ⋯ |
L(s) = 1 | + (−0.280 + 0.959i)2-s + (0.321 − 0.556i)3-s + (−0.842 − 0.539i)4-s + (0.387 − 0.223i)5-s + (0.443 + 0.464i)6-s + (0.878 + 0.477i)7-s + (0.753 − 0.656i)8-s + (0.293 + 0.508i)9-s + (0.105 + 0.434i)10-s + (0.447 + 0.258i)11-s + (−0.570 + 0.295i)12-s − 0.682i·13-s + (−0.705 + 0.708i)14-s − 0.287i·15-s + (0.418 + 0.908i)16-s + (−1.30 − 0.754i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.808 - 0.589i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.808 - 0.589i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.05820 + 0.344754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05820 + 0.344754i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.397 - 1.35i)T \) |
| 5 | \( 1 + (-0.866 + 0.5i)T \) |
| 7 | \( 1 + (-2.32 - 1.26i)T \) |
good | 3 | \( 1 + (-0.556 + 0.963i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.48 - 0.856i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 2.45iT - 13T^{2} \) |
| 17 | \( 1 + (5.38 + 3.10i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.108 + 0.187i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.68 - 3.28i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2.47T + 29T^{2} \) |
| 31 | \( 1 + (-0.0819 + 0.141i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.84 + 6.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 8.34iT - 41T^{2} \) |
| 43 | \( 1 + 1.89iT - 43T^{2} \) |
| 47 | \( 1 + (5.85 + 10.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.51 - 11.2i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (2.14 - 3.71i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.06 + 3.50i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-4.48 - 2.58i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.04iT - 71T^{2} \) |
| 73 | \( 1 + (-6.59 - 3.80i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (13.8 - 7.97i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 5.47T + 83T^{2} \) |
| 89 | \( 1 + (1.54 - 0.891i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10.5iT - 97T^{2} \) |
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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
−13.54705159223861220492070057088, −12.57773711222837923189086018371, −11.16474866322077991680426651582, −9.862900881530105992186448484012, −8.791105370767636447221567913708, −7.944052019634450020508749169149, −6.98464014554274045113741632562, −5.63901090465639230813172637704, −4.55867188401360444645504517767, −1.87632345874529691977832991857,
1.87231100127896439165146749349, 3.75022907698074578324449757267, 4.62077543133401738587734551703, 6.60363428031995614827702303576, 8.223237242789068949752966136592, 9.091649302175423718637739787505, 10.07157858730768528591635111697, 10.92255397275060662167953625312, 11.81168063860350969821575061823, 12.99648366490729174161297706286