| L(s) = 1 | + (−1.19 − 2.34i)2-s + (−0.686 + 1.59i)3-s + (−2.87 + 3.96i)4-s + (−0.296 − 1.87i)5-s + (4.53 − 0.290i)6-s + (0.760 − 0.649i)7-s + (7.51 + 1.19i)8-s + (−2.05 − 2.18i)9-s + (−4.03 + 2.92i)10-s + (3.93 + 2.41i)11-s + (−4.32 − 7.29i)12-s + (−5.22 + 0.411i)13-s + (−2.42 − 1.00i)14-s + (3.18 + 0.814i)15-s + (−3.14 − 9.69i)16-s + (−5.76 + 1.38i)17-s + ⋯ |
| L(s) = 1 | + (−0.843 − 1.65i)2-s + (−0.396 + 0.918i)3-s + (−1.43 + 1.98i)4-s + (−0.132 − 0.838i)5-s + (1.85 − 0.118i)6-s + (0.287 − 0.245i)7-s + (2.65 + 0.420i)8-s + (−0.686 − 0.727i)9-s + (−1.27 + 0.926i)10-s + (1.18 + 0.727i)11-s + (−1.24 − 2.10i)12-s + (−1.44 + 0.114i)13-s + (−0.648 − 0.268i)14-s + (0.822 + 0.210i)15-s + (−0.787 − 2.42i)16-s + (−1.39 + 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 861 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.370 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.580399 - 0.393421i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.580399 - 0.393421i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.686 - 1.59i)T \) |
| 7 | \( 1 + (-0.760 + 0.649i)T \) |
| 41 | \( 1 + (-4.08 + 4.92i)T \) |
| good | 2 | \( 1 + (1.19 + 2.34i)T + (-1.17 + 1.61i)T^{2} \) |
| 5 | \( 1 + (0.296 + 1.87i)T + (-4.75 + 1.54i)T^{2} \) |
| 11 | \( 1 + (-3.93 - 2.41i)T + (4.99 + 9.80i)T^{2} \) |
| 13 | \( 1 + (5.22 - 0.411i)T + (12.8 - 2.03i)T^{2} \) |
| 17 | \( 1 + (5.76 - 1.38i)T + (15.1 - 7.71i)T^{2} \) |
| 19 | \( 1 + (-3.75 - 0.295i)T + (18.7 + 2.97i)T^{2} \) |
| 23 | \( 1 + (-0.0189 + 0.0582i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.279 - 0.0670i)T + (25.8 + 13.1i)T^{2} \) |
| 31 | \( 1 + (-4.07 - 5.60i)T + (-9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.86 - 4.99i)T + (11.4 + 35.1i)T^{2} \) |
| 43 | \( 1 + (2.58 - 1.31i)T + (25.2 - 34.7i)T^{2} \) |
| 47 | \( 1 + (-2.07 + 2.42i)T + (-7.35 - 46.4i)T^{2} \) |
| 53 | \( 1 + (1.30 - 5.43i)T + (-47.2 - 24.0i)T^{2} \) |
| 59 | \( 1 + (-0.321 - 0.104i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (-5.54 + 10.8i)T + (-35.8 - 49.3i)T^{2} \) |
| 67 | \( 1 + (-8.53 + 5.23i)T + (30.4 - 59.6i)T^{2} \) |
| 71 | \( 1 + (-6.18 + 10.0i)T + (-32.2 - 63.2i)T^{2} \) |
| 73 | \( 1 + (8.67 + 8.67i)T + 73iT^{2} \) |
| 79 | \( 1 + (-13.7 + 5.69i)T + (55.8 - 55.8i)T^{2} \) |
| 83 | \( 1 - 11.1iT - 83T^{2} \) |
| 89 | \( 1 + (-5.51 - 6.45i)T + (-13.9 + 87.9i)T^{2} \) |
| 97 | \( 1 + (5.49 + 8.96i)T + (-44.0 + 86.4i)T^{2} \) |
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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
−9.964366929606838603009713811113, −9.363113770269262981765768410303, −8.917913733975404718582152815086, −7.944895842338116209521769757275, −6.70239833579084248120315241269, −4.72486528983720711143587202760, −4.63297967149268685330171268499, −3.51179327632266264451008526575, −2.17989309555334218767124864939, −0.818817915676523738486962719177,
0.75382269965114373348704549405, 2.49471833206467492189058960599, 4.53053281562399504481906851207, 5.56638548817931760654816254706, 6.35387109180294286575130029950, 6.99056252319731312449202844877, 7.50698238574947839935170188985, 8.388720935696519001141219930955, 9.198313661743922451534987079222, 9.995767011400639063469087058048
