L(s) = 1 | + (0.415 + 0.571i)2-s + (0.463 − 1.42i)4-s + (−2.26 − 0.734i)5-s + (−4.85 + 0.768i)7-s + (2.35 − 0.764i)8-s + (−0.519 − 1.59i)10-s + (−1.51 − 0.773i)11-s + (0.621 − 3.92i)13-s + (−2.45 − 2.45i)14-s + (−1.01 − 0.736i)16-s + (1.24 − 2.44i)17-s + (−0.150 − 0.953i)19-s + (−2.09 + 2.88i)20-s + (−0.188 − 1.18i)22-s + (−5.46 + 3.97i)23-s + ⋯ |
L(s) = 1 | + (0.293 + 0.404i)2-s + (0.231 − 0.713i)4-s + (−1.01 − 0.328i)5-s + (−1.83 + 0.290i)7-s + (0.831 − 0.270i)8-s + (−0.164 − 0.505i)10-s + (−0.457 − 0.233i)11-s + (0.172 − 1.08i)13-s + (−0.656 − 0.656i)14-s + (−0.253 − 0.184i)16-s + (0.302 − 0.593i)17-s + (−0.0346 − 0.218i)19-s + (−0.469 + 0.645i)20-s + (−0.0401 − 0.253i)22-s + (−1.13 + 0.828i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.509 + 0.860i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.340393 - 0.596860i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.340393 - 0.596860i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 41 | \( 1 + (-4.01 - 4.98i)T \) |
good | 2 | \( 1 + (-0.415 - 0.571i)T + (-0.618 + 1.90i)T^{2} \) |
| 5 | \( 1 + (2.26 + 0.734i)T + (4.04 + 2.93i)T^{2} \) |
| 7 | \( 1 + (4.85 - 0.768i)T + (6.65 - 2.16i)T^{2} \) |
| 11 | \( 1 + (1.51 + 0.773i)T + (6.46 + 8.89i)T^{2} \) |
| 13 | \( 1 + (-0.621 + 3.92i)T + (-12.3 - 4.01i)T^{2} \) |
| 17 | \( 1 + (-1.24 + 2.44i)T + (-9.99 - 13.7i)T^{2} \) |
| 19 | \( 1 + (0.150 + 0.953i)T + (-18.0 + 5.87i)T^{2} \) |
| 23 | \( 1 + (5.46 - 3.97i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (0.230 + 0.451i)T + (-17.0 + 23.4i)T^{2} \) |
| 31 | \( 1 + (-0.182 - 0.561i)T + (-25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-1.31 + 4.04i)T + (-29.9 - 21.7i)T^{2} \) |
| 43 | \( 1 + (-3.16 - 4.35i)T + (-13.2 + 40.8i)T^{2} \) |
| 47 | \( 1 + (-4.96 - 0.786i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (3.46 + 6.79i)T + (-31.1 + 42.8i)T^{2} \) |
| 59 | \( 1 + (-6.81 + 4.95i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (-0.408 + 0.562i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (3.63 - 1.85i)T + (39.3 - 54.2i)T^{2} \) |
| 71 | \( 1 + (6.47 + 3.30i)T + (41.7 + 57.4i)T^{2} \) |
| 73 | \( 1 - 9.72iT - 73T^{2} \) |
| 79 | \( 1 + (-6.15 + 6.15i)T - 79iT^{2} \) |
| 83 | \( 1 + 4.81T + 83T^{2} \) |
| 89 | \( 1 + (-3.63 + 0.576i)T + (84.6 - 27.5i)T^{2} \) |
| 97 | \( 1 + (3.63 - 1.85i)T + (57.0 - 78.4i)T^{2} \) |
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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
−11.08587770553440206485723228946, −10.06259097022073431009747172264, −9.454856403150976956842581468185, −8.080855222803458310153190678988, −7.23798436094750895755365319985, −6.13508016143827074886510397801, −5.45724276191487069139013753032, −4.02091837013041738112392465132, −2.88763237629612212177702375536, −0.40051935045085191860739942067,
2.53615902593249571126943920485, 3.69032264764482665728607793345, 4.16184073532518492088964087124, 6.15626204183056957367183001026, 7.05266250714025139775273146998, 7.80053729367713795142683793441, 8.941969046624296333437891221816, 10.12092645075650990871401206218, 10.84349098891395745177687400232, 12.01366746582292075070933062223