L(s) = 1 | + (0.5 − 0.866i)2-s + (2.59 − 1.5i)3-s + (−0.499 − 0.866i)4-s − 3i·6-s + (−1.73 + 2i)7-s − 0.999·8-s + (3 − 5.19i)9-s + (−2.59 + 1.5i)11-s + (−2.59 − 1.50i)12-s + 5·13-s + (0.866 + 2.5i)14-s + (−0.5 + 0.866i)16-s + (−3.96 + 1.13i)17-s + (−3 − 5.19i)18-s + (−2 + 3.46i)19-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (1.49 − 0.866i)3-s + (−0.249 − 0.433i)4-s − 1.22i·6-s + (−0.654 + 0.755i)7-s − 0.353·8-s + (1 − 1.73i)9-s + (−0.783 + 0.452i)11-s + (−0.749 − 0.433i)12-s + 1.38·13-s + (0.231 + 0.668i)14-s + (−0.125 + 0.216i)16-s + (−0.961 + 0.275i)17-s + (−0.707 − 1.22i)18-s + (−0.458 + 0.794i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 238 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.151 + 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.56106 - 1.34032i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.56106 - 1.34032i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 + (1.73 - 2i)T \) |
| 17 | \( 1 + (3.96 - 1.13i)T \) |
good | 3 | \( 1 + (-2.59 + 1.5i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.59 - 1.5i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 19 | \( 1 + (2 - 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.46 - 2i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 + (-3.46 + 2i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.92 + 4i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4iT - 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 + (2 - 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6 + 10.3i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-8 - 13.8i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + iT - 71T^{2} \) |
| 73 | \( 1 + (6.92 - 4i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.06 - 3.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 16T + 83T^{2} \) |
| 89 | \( 1 + (4.5 - 7.79i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 8iT - 97T^{2} \) |
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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
−12.47142970934358923879715210482, −11.03511312362808185991962702412, −9.866697577356648512265673233499, −8.840872255957439781817503957091, −8.308795883507136527433395593556, −6.95067880007648265560145080139, −5.86475833711548308583325196366, −3.97350140959636375341568508133, −2.88194820610196502573256478848, −1.87072485956510633802368904664,
2.84326857448387940484128249192, 3.76605421545594811879256021514, 4.76011602936817966651137700162, 6.38956897147667516533394771779, 7.54996769182271527692556209778, 8.564029696573533914418462573057, 9.139395021082871822422959133033, 10.30291718521800745403703180384, 11.10851031989385042745150611802, 13.08927072110243034878179995581