L(s) = 1 | + (−0.145 − 0.251i)2-s + (0.957 − 1.65i)4-s + (0.5 + 0.866i)5-s − 0.486·7-s − 1.13·8-s + (0.145 − 0.251i)10-s − 5.34·11-s + (1.24 − 2.15i)13-s + (0.0707 + 0.122i)14-s + (−1.75 − 3.03i)16-s + (−1.70 − 2.95i)17-s + (−2.46 − 3.59i)19-s + 1.91·20-s + (0.777 + 1.34i)22-s + (3.20 − 5.55i)23-s + ⋯ |
L(s) = 1 | + (−0.102 − 0.178i)2-s + (0.478 − 0.829i)4-s + (0.223 + 0.387i)5-s − 0.183·7-s − 0.402·8-s + (0.0459 − 0.0796i)10-s − 1.61·11-s + (0.344 − 0.597i)13-s + (0.0189 + 0.0327i)14-s + (−0.437 − 0.757i)16-s + (−0.414 − 0.717i)17-s + (−0.565 − 0.825i)19-s + 0.428·20-s + (0.165 + 0.287i)22-s + (0.668 − 1.15i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 855 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.612 + 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.492122 - 1.00435i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.492122 - 1.00435i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (2.46 + 3.59i)T \) |
good | 2 | \( 1 + (0.145 + 0.251i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + 0.486T + 7T^{2} \) |
| 11 | \( 1 + 5.34T + 11T^{2} \) |
| 13 | \( 1 + (-1.24 + 2.15i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (1.70 + 2.95i)T + (-8.5 + 14.7i)T^{2} \) |
| 23 | \( 1 + (-3.20 + 5.55i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.38 + 2.39i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 4.83T + 31T^{2} \) |
| 37 | \( 1 + 6.31T + 37T^{2} \) |
| 41 | \( 1 + (-1.29 - 2.23i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.24 - 2.15i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-5.55 + 9.62i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.49 + 2.59i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.36 - 11.0i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.92 - 10.2i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.58 + 13.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (4.96 + 8.59i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-5.50 - 9.53i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.06 + 7.04i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 5.86T + 83T^{2} \) |
| 89 | \( 1 + (3.25 - 5.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−10.24526682669309230737741205681, −9.154288669004501397091602183864, −8.241902538173743186722697755733, −7.13554180677777775331383644364, −6.44799121110183099749041722632, −5.47675882515386488474807177059, −4.72053066229396410016320458901, −2.91451809567342662777303383941, −2.36363389935450300142264967125, −0.51779915152568459489364373067,
1.88217393351276551773856321883, 3.01593181329600836600305933497, 4.08437095595011331828871110763, 5.27552217310230911694476247328, 6.22518783245651658566675423442, 7.12431786674917558533953253646, 8.056501244426341446235547071878, 8.547969898370372098738857458045, 9.559002939347053813812887450704, 10.58539434245057771840490817271