L(s) = 1 | + (0.116 − 1.40i)2-s + (−1.97 − 0.327i)4-s − 2.99·5-s − i·7-s + (−0.690 + 2.74i)8-s + (−0.347 + 4.22i)10-s − 5.36i·11-s − 4.55i·13-s + (−1.40 − 0.116i)14-s + (3.78 + 1.29i)16-s − 0.958i·17-s + 0.532·19-s + (5.91 + 0.980i)20-s + (−7.55 − 0.622i)22-s − 2.78·23-s + ⋯ |
L(s) = 1 | + (0.0820 − 0.996i)2-s + (−0.986 − 0.163i)4-s − 1.34·5-s − 0.377i·7-s + (−0.243 + 0.969i)8-s + (−0.109 + 1.33i)10-s − 1.61i·11-s − 1.26i·13-s + (−0.376 − 0.0310i)14-s + (0.946 + 0.322i)16-s − 0.232i·17-s + 0.122·19-s + (1.32 + 0.219i)20-s + (−1.61 − 0.132i)22-s − 0.581·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.243 - 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.1273554140\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1273554140\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.116 + 1.40i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
good | 5 | \( 1 + 2.99T + 5T^{2} \) |
| 11 | \( 1 + 5.36iT - 11T^{2} \) |
| 13 | \( 1 + 4.55iT - 13T^{2} \) |
| 17 | \( 1 + 0.958iT - 17T^{2} \) |
| 19 | \( 1 - 0.532T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 + 5.15T + 29T^{2} \) |
| 31 | \( 1 - 4.66iT - 31T^{2} \) |
| 37 | \( 1 - 6.10iT - 37T^{2} \) |
| 41 | \( 1 - 12.3iT - 41T^{2} \) |
| 43 | \( 1 - 4.45T + 43T^{2} \) |
| 47 | \( 1 + 1.40T + 47T^{2} \) |
| 53 | \( 1 + 9.57T + 53T^{2} \) |
| 59 | \( 1 + 0.0356iT - 59T^{2} \) |
| 61 | \( 1 - 5.13iT - 61T^{2} \) |
| 67 | \( 1 - 8.59T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 6.50T + 73T^{2} \) |
| 79 | \( 1 + 1.90iT - 79T^{2} \) |
| 83 | \( 1 + 3.13iT - 83T^{2} \) |
| 89 | \( 1 + 5.91iT - 89T^{2} \) |
| 97 | \( 1 + 16.0T + 97T^{2} \) |
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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
−8.769694845016358531211056975123, −8.106350503921454740986785172115, −7.66393400928978624320786916118, −6.22751136542701213754443611940, −5.29666112721511955965769651524, −4.33700257728420133388116771438, −3.40146788719803295938740853731, −3.01392310064659763432317876270, −1.10573984035904567956820447903, −0.05828359353517081406244820523,
2.04598539590204950123224612665, 3.81610135791905604237418749524, 4.19907043414613103657883348790, 5.09413301846709715690564755350, 6.16675843078660471289051065736, 7.17324306745427214461980905865, 7.46811886248516428617186059750, 8.314063536697571966856088365351, 9.223328281211721270062338266095, 9.692509494510790862511202302442