L(s) = 1 | + (−1.39 − 2.40i)2-s + (−0.5 + 0.866i)3-s + (−2.86 + 4.96i)4-s + (−0.412 − 0.715i)5-s + 2.78·6-s + (2.63 + 0.257i)7-s + 10.3·8-s + (−0.499 − 0.866i)9-s + (−1.14 + 1.98i)10-s + (0.5 − 0.866i)11-s + (−2.86 − 4.96i)12-s − 0.296·13-s + (−3.04 − 6.70i)14-s + 0.825·15-s + (−8.72 − 15.1i)16-s + (3.34 − 5.79i)17-s + ⋯ |
L(s) = 1 | + (−0.983 − 1.70i)2-s + (−0.288 + 0.499i)3-s + (−1.43 + 2.48i)4-s + (−0.184 − 0.319i)5-s + 1.13·6-s + (0.995 + 0.0971i)7-s + 3.67·8-s + (−0.166 − 0.288i)9-s + (−0.363 + 0.629i)10-s + (0.150 − 0.261i)11-s + (−0.828 − 1.43i)12-s − 0.0823·13-s + (−0.813 − 1.79i)14-s + 0.213·15-s + (−2.18 − 3.77i)16-s + (0.811 − 1.40i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.160 + 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.445395 - 0.523402i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.445395 - 0.523402i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.63 - 0.257i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (1.39 + 2.40i)T + (-1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (0.412 + 0.715i)T + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 + 0.296T + 13T^{2} \) |
| 17 | \( 1 + (-3.34 + 5.79i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.41 - 2.45i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.98 - 3.43i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 0.484T + 29T^{2} \) |
| 31 | \( 1 + (-3.66 + 6.34i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.86 - 4.96i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 0.645T + 41T^{2} \) |
| 43 | \( 1 - 6.43T + 43T^{2} \) |
| 47 | \( 1 + (3.86 + 6.70i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (3.55 - 6.16i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.578 + 1.00i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.63 + 4.56i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.50 - 2.61i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 3.58T + 71T^{2} \) |
| 73 | \( 1 + (8.01 - 13.8i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2.16 - 3.74i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.37T + 83T^{2} \) |
| 89 | \( 1 + (6.08 + 10.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−11.68267300445339053664483930586, −11.09397033937807838378265122687, −10.02945340423348937771955248873, −9.348461615269938286962487492420, −8.342802264583581183374375800703, −7.55708143579859035272896148622, −5.13414987572860930729847082385, −4.13062593095988891962758771799, −2.78971114026863799858056733079, −1.05184918873545224089879531193,
1.33452186319518123630249285577, 4.59895188212412298440741821523, 5.62583662310436766154358543851, 6.66298002830413542276300542794, 7.51925726233525389224036345384, 8.202438772416156572894331406412, 9.149112959766927191402995598404, 10.42256420652522051984083745854, 11.07278821587219127500822308392, 12.62531066176668062747804583238