L(s) = 1 | + (0.757 + 1.31i)2-s + (−0.419 + 1.68i)3-s + (−0.147 + 0.254i)4-s + (−1.42 + 1.72i)5-s + (−2.52 + 0.722i)6-s + (−0.0753 − 2.64i)7-s + 2.58·8-s + (−2.64 − 1.41i)9-s + (−3.33 − 0.570i)10-s + (1.86 + 1.07i)11-s + (−0.366 − 0.354i)12-s + 3.48·13-s + (3.41 − 2.10i)14-s + (−2.29 − 3.12i)15-s + (2.25 + 3.89i)16-s + (−3.09 − 1.78i)17-s + ⋯ |
L(s) = 1 | + (0.535 + 0.927i)2-s + (−0.242 + 0.970i)3-s + (−0.0735 + 0.127i)4-s + (−0.638 + 0.769i)5-s + (−1.02 + 0.294i)6-s + (−0.0284 − 0.999i)7-s + 0.913·8-s + (−0.882 − 0.470i)9-s + (−1.05 − 0.180i)10-s + (0.560 + 0.323i)11-s + (−0.105 − 0.102i)12-s + 0.965·13-s + (0.911 − 0.561i)14-s + (−0.591 − 0.806i)15-s + (0.562 + 0.974i)16-s + (−0.751 − 0.433i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.261 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.735561 + 0.961647i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.735561 + 0.961647i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.419 - 1.68i)T \) |
| 5 | \( 1 + (1.42 - 1.72i)T \) |
| 7 | \( 1 + (0.0753 + 2.64i)T \) |
good | 2 | \( 1 + (-0.757 - 1.31i)T + (-1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (-1.86 - 1.07i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.48T + 13T^{2} \) |
| 17 | \( 1 + (3.09 + 1.78i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.05 - 0.611i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.757 + 1.31i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5.95iT - 29T^{2} \) |
| 31 | \( 1 + (-2.75 - 1.58i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.75 - 3.90i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 + 2.99iT - 43T^{2} \) |
| 47 | \( 1 + (-5.28 + 3.05i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.61 - 9.72i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.08 - 1.87i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.94 - 1.69i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-8.93 - 5.15i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3iT - 71T^{2} \) |
| 73 | \( 1 + (-3.42 + 5.93i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.941 - 1.63i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 9.10iT - 83T^{2} \) |
| 89 | \( 1 + (-0.889 - 1.54i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 1.32T + 97T^{2} \) |
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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
−14.21945808516040609233068643662, −13.63609691930384272700846779414, −11.75484029226745409338962481126, −10.80004749271291689692802515561, −10.11514558254457220093782905531, −8.412970021436165536612552184850, −7.03129621623134958336911868651, −6.24496185939555542196640911353, −4.60923317961375417863802379596, −3.74172143742715023096064998570,
1.72895176123632464323396590503, 3.50080522506720314290781185096, 5.12106293113770080819780605319, 6.57463180238235290826798478843, 8.117546375020559407355254157614, 8.894349241575556758415959457623, 10.95805401438911578497975423546, 11.64164414696031443515383497984, 12.40519057789801086682967006149, 13.04492199077173298567671403938