L(s) = 1 | + (0.5 − 0.866i)3-s + (−0.5 − 0.866i)5-s + (−2 + 1.73i)7-s + (−0.499 − 0.866i)9-s + (−0.5 + 0.866i)11-s + 7·13-s − 0.999·15-s + (2 − 3.46i)17-s + (0.5 + 0.866i)19-s + (0.499 + 2.59i)21-s + (0.5 + 0.866i)23-s + (−0.499 + 0.866i)25-s − 0.999·27-s − 8·29-s + (3 − 5.19i)31-s + ⋯ |
L(s) = 1 | + (0.288 − 0.499i)3-s + (−0.223 − 0.387i)5-s + (−0.755 + 0.654i)7-s + (−0.166 − 0.288i)9-s + (−0.150 + 0.261i)11-s + 1.94·13-s − 0.258·15-s + (0.485 − 0.840i)17-s + (0.114 + 0.198i)19-s + (0.109 + 0.566i)21-s + (0.104 + 0.180i)23-s + (−0.0999 + 0.173i)25-s − 0.192·27-s − 1.48·29-s + (0.538 − 0.933i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.737288012\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.737288012\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.5 + 0.866i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (2 - 1.73i)T \) |
good | 11 | \( 1 + (0.5 - 0.866i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 7T + 13T^{2} \) |
| 17 | \( 1 + (-2 + 3.46i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + (-3 + 5.19i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.5 - 2.59i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + (1.5 + 2.59i)T + (-23.5 + 40.7i)T^{2} \) |
| 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 | \( 1 + (-2 - 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6 + 10.3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 14T + 71T^{2} \) |
| 73 | \( 1 + (-7 + 12.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 12T + 83T^{2} \) |
| 89 | \( 1 + (-1 - 1.73i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 16T + 97T^{2} \) |
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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
−9.326571567935796344720809965382, −8.366840344029233311283527653321, −7.81427208947639451250651817126, −6.79885697797360055294478179234, −6.01366793770760016084888204001, −5.36131440446448228118158327346, −3.97882251719869522592019268861, −3.25994944754442350787060715362, −2.11437616740694529608020639293, −0.796668093031243616817377443772,
1.10077133267677927312309517926, 2.74554594674542759618937612781, 3.77103974420012038448572195734, 3.98760570273129490453574010863, 5.54673462980279822339860205102, 6.20620881299514860608967374891, 7.07737911629179315651648548378, 7.992089930100106946891583691678, 8.687309517365665087144115677869, 9.480497401198091138442506620414