| L(s) = 1 | + (−1.11 + 0.869i)2-s + (0.540 − 0.841i)3-s + (0.489 − 1.93i)4-s + (−0.756 − 0.345i)5-s + (0.127 + 1.40i)6-s + (−3.66 − 1.07i)7-s + (1.13 + 2.58i)8-s + (−0.415 − 0.909i)9-s + (1.14 − 0.272i)10-s + (2.56 + 2.22i)11-s + (−1.36 − 1.46i)12-s + (0.635 + 2.16i)13-s + (5.02 − 1.98i)14-s + (−0.699 + 0.449i)15-s + (−3.52 − 1.89i)16-s + (−1.05 − 7.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.788 + 0.614i)2-s + (0.312 − 0.485i)3-s + (0.244 − 0.969i)4-s + (−0.338 − 0.154i)5-s + (0.0522 + 0.574i)6-s + (−1.38 − 0.406i)7-s + (0.402 + 0.915i)8-s + (−0.138 − 0.303i)9-s + (0.361 − 0.0860i)10-s + (0.772 + 0.669i)11-s + (−0.394 − 0.421i)12-s + (0.176 + 0.600i)13-s + (1.34 − 0.530i)14-s + (−0.180 + 0.116i)15-s + (−0.880 − 0.474i)16-s + (−0.256 − 1.78i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.943 + 0.330i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0274468 - 0.161428i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0274468 - 0.161428i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.11 - 0.869i)T \) |
| 3 | \( 1 + (-0.540 + 0.841i)T \) |
| 23 | \( 1 + (-2.22 - 4.25i)T \) |
| good | 5 | \( 1 + (0.756 + 0.345i)T + (3.27 + 3.77i)T^{2} \) |
| 7 | \( 1 + (3.66 + 1.07i)T + (5.88 + 3.78i)T^{2} \) |
| 11 | \( 1 + (-2.56 - 2.22i)T + (1.56 + 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.635 - 2.16i)T + (-10.9 + 7.02i)T^{2} \) |
| 17 | \( 1 + (1.05 + 7.35i)T + (-16.3 + 4.78i)T^{2} \) |
| 19 | \( 1 + (5.92 + 0.852i)T + (18.2 + 5.35i)T^{2} \) |
| 29 | \( 1 + (6.91 - 0.994i)T + (27.8 - 8.17i)T^{2} \) |
| 31 | \( 1 + (5.48 - 3.52i)T + (12.8 - 28.1i)T^{2} \) |
| 37 | \( 1 + (6.41 - 2.92i)T + (24.2 - 27.9i)T^{2} \) |
| 41 | \( 1 + (2.34 - 5.13i)T + (-26.8 - 30.9i)T^{2} \) |
| 43 | \( 1 + (0.890 - 1.38i)T + (-17.8 - 39.1i)T^{2} \) |
| 47 | \( 1 + 6.67T + 47T^{2} \) |
| 53 | \( 1 + (-2.73 + 9.32i)T + (-44.5 - 28.6i)T^{2} \) |
| 59 | \( 1 + (-1.99 - 6.78i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (2.08 + 3.23i)T + (-25.3 + 55.4i)T^{2} \) |
| 67 | \( 1 + (0.672 - 0.582i)T + (9.53 - 66.3i)T^{2} \) |
| 71 | \( 1 + (1.95 + 2.25i)T + (-10.1 + 70.2i)T^{2} \) |
| 73 | \( 1 + (-1.68 + 11.7i)T + (-70.0 - 20.5i)T^{2} \) |
| 79 | \( 1 + (-5.10 + 1.50i)T + (66.4 - 42.7i)T^{2} \) |
| 83 | \( 1 + (-4.72 + 2.15i)T + (54.3 - 62.7i)T^{2} \) |
| 89 | \( 1 + (-14.8 - 9.51i)T + (36.9 + 80.9i)T^{2} \) |
| 97 | \( 1 + (-7.16 + 15.6i)T + (-63.5 - 73.3i)T^{2} \) |
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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
−10.00773063323946429644759865465, −9.333435053175560971622945386046, −8.794926119405318603141520678041, −7.48186350041764509703641658235, −6.90022955446900930363425197193, −6.32777885698611988793037788845, −4.83978703221371927153457672832, −3.48551447161175791740760665456, −1.88579143498977374376659751866, −0.11014190068710026401414286453,
2.12589235833445762991116523208, 3.59463874812330681757828785461, 3.79362339107439562477020009289, 5.90300787420335770565734857470, 6.70946675841753591431705975368, 7.973186866022070225963783580287, 8.837474267199460770928912098315, 9.274147229858789215113052145965, 10.46512681194055441887890492055, 10.77027314689130024234899910872
