L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (1.20 + 2.09i)5-s + (1.62 − 2.09i)7-s − 0.999·8-s + (−1.20 + 2.09i)10-s + (0.5 − 0.866i)11-s − 6.82·13-s + (2.62 + 0.358i)14-s + (−0.5 − 0.866i)16-s + (−3.91 + 6.77i)17-s + (2.41 + 4.18i)19-s − 2.41·20-s + 0.999·22-s + (2.62 + 4.54i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.539 + 0.935i)5-s + (0.612 − 0.790i)7-s − 0.353·8-s + (−0.381 + 0.661i)10-s + (0.150 − 0.261i)11-s − 1.89·13-s + (0.700 + 0.0958i)14-s + (−0.125 − 0.216i)16-s + (−0.949 + 1.64i)17-s + (0.553 + 0.959i)19-s − 0.539·20-s + 0.213·22-s + (0.546 + 0.946i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.749 - 0.661i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.793853276\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.793853276\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-1.62 + 2.09i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-1.20 - 2.09i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 6.82T + 13T^{2} \) |
| 17 | \( 1 + (3.91 - 6.77i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.41 - 4.18i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 4.54i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + (5.24 - 9.08i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.828 - 1.43i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6.65T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + (0.378 + 0.655i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.41 - 9.37i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.82 + 6.63i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.20 - 3.82i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.74 - 3.01i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 9.31T + 71T^{2} \) |
| 73 | \( 1 + (0.414 - 0.717i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (6.62 + 11.4i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 4.17T + 83T^{2} \) |
| 89 | \( 1 + (2.24 + 3.88i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 15.8T + 97T^{2} \) |
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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
−10.00317132171155064411588538507, −9.025638923056664694952054534158, −7.996850512832606669151519144619, −7.31423455827977500751907899775, −6.72154936986498187605117110998, −5.81244741522650276625272603062, −4.91916754104161788336795775564, −4.00041023137910432940469268598, −2.95874300717292081382872984800, −1.72812060966195884419057826533,
0.62620629884395417407779320090, 2.24456830159863625944498970796, 2.61357252844115598472692173870, 4.49925639639893120650247363208, 4.92873320033882545165258287102, 5.49057891625365421758971751259, 6.80476004567841863493852147232, 7.64995701679328947565621727535, 8.852374529401760582606701458086, 9.382659765370747703995964782898