L(s) = 1 | + (0.5 + 0.866i)2-s + (−0.499 + 0.866i)4-s + (0.707 + 1.22i)5-s + (−1.62 + 2.09i)7-s − 0.999·8-s + (−0.707 + 1.22i)10-s + (−1.20 + 2.09i)11-s + 13-s + (−2.62 − 0.358i)14-s + (−0.5 − 0.866i)16-s + (−1.62 + 2.80i)17-s + (−1.5 − 2.59i)19-s − 1.41·20-s − 2.41·22-s + (2.53 + 4.39i)23-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.316 + 0.547i)5-s + (−0.612 + 0.790i)7-s − 0.353·8-s + (−0.223 + 0.387i)10-s + (−0.363 + 0.630i)11-s + 0.277·13-s + (−0.700 − 0.0958i)14-s + (−0.125 − 0.216i)16-s + (−0.393 + 0.681i)17-s + (−0.344 − 0.596i)19-s − 0.316·20-s − 0.514·22-s + (0.528 + 0.915i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.947 + 0.318i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.118058393\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.118058393\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.62 - 2.09i)T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + (-0.707 - 1.22i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.20 - 2.09i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (1.62 - 2.80i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 + 2.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.53 - 4.39i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 + (0.585 - 1.01i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.53 + 7.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 7.41T + 41T^{2} \) |
| 43 | \( 1 + 0.828T + 43T^{2} \) |
| 47 | \( 1 + (-4.5 - 7.79i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.15 - 8.93i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.20 + 9.01i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (1.62 + 2.80i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.67 + 2.89i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - T + 71T^{2} \) |
| 73 | \( 1 + (6.53 - 11.3i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-4.65 - 8.06i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 3.65T + 83T^{2} \) |
| 89 | \( 1 + (-1.12 - 1.94i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 11.8T + 97T^{2} \) |
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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
−9.603717602076758592938774526110, −9.033198902966712115991492246735, −8.192280651350849961362858050970, −7.18473419213227798188453818120, −6.61178060205476767957497430326, −5.81511888680953991774262337483, −5.10346289806759034771033592057, −3.96860559670531849130754250833, −2.99112764699686567252400092114, −2.03631405010554828785559619864,
0.36279140917537326071244194323, 1.61896209378890804740382336567, 2.95168054001931380070451086870, 3.73524013890505224241054850894, 4.73576611442128051539809412435, 5.49761287245705842699161019781, 6.44682306326228412609456769538, 7.21990660586359372672366292104, 8.417738030434338091985988554718, 8.990230155442161079243254037104