L(s) = 1 | + (0.576 + 1.29i)2-s + 2.50i·3-s + (−1.33 + 1.48i)4-s + (0.639 + 2.14i)5-s + (−3.23 + 1.44i)6-s + (−2.69 − 0.867i)8-s − 3.25·9-s + (−2.39 + 2.06i)10-s − 2.25i·11-s + (−3.72 − 3.34i)12-s − 5.96·13-s + (−5.35 + 1.60i)15-s + (−0.430 − 3.97i)16-s + 2.00·17-s + (−1.87 − 4.20i)18-s + 7.81·19-s + ⋯ |
L(s) = 1 | + (0.407 + 0.913i)2-s + 1.44i·3-s + (−0.667 + 0.744i)4-s + (0.286 + 0.958i)5-s + (−1.31 + 0.588i)6-s + (−0.951 − 0.306i)8-s − 1.08·9-s + (−0.758 + 0.651i)10-s − 0.678i·11-s + (−1.07 − 0.964i)12-s − 1.65·13-s + (−1.38 + 0.413i)15-s + (−0.107 − 0.994i)16-s + 0.486·17-s + (−0.442 − 0.991i)18-s + 1.79·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.280 + 0.959i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.807714 - 1.07716i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.807714 - 1.07716i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.576 - 1.29i)T \) |
| 5 | \( 1 + (-0.639 - 2.14i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2.50iT - 3T^{2} \) |
| 11 | \( 1 + 2.25iT - 11T^{2} \) |
| 13 | \( 1 + 5.96T + 13T^{2} \) |
| 17 | \( 1 - 2.00T + 17T^{2} \) |
| 19 | \( 1 - 7.81T + 19T^{2} \) |
| 23 | \( 1 + 2.99T + 23T^{2} \) |
| 29 | \( 1 + 4.87T + 29T^{2} \) |
| 31 | \( 1 + 1.49T + 31T^{2} \) |
| 37 | \( 1 - 4.78iT - 37T^{2} \) |
| 41 | \( 1 - 8.82iT - 41T^{2} \) |
| 43 | \( 1 - 1.12T + 43T^{2} \) |
| 47 | \( 1 - 9.56iT - 47T^{2} \) |
| 53 | \( 1 - 7.06iT - 53T^{2} \) |
| 59 | \( 1 - 11.4T + 59T^{2} \) |
| 61 | \( 1 + 1.21iT - 61T^{2} \) |
| 67 | \( 1 - 1.11T + 67T^{2} \) |
| 71 | \( 1 + 8.40iT - 71T^{2} \) |
| 73 | \( 1 + 5.88T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 11.1iT - 83T^{2} \) |
| 89 | \( 1 - 4.57iT - 89T^{2} \) |
| 97 | \( 1 + 4.62T + 97T^{2} \) |
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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
−10.25083842959889645503787141311, −9.693615767361617856722752099775, −9.192126905639837662091902395947, −7.82066495949770049506451921185, −7.31054904044431464460121764958, −6.09139103035394457889454536081, −5.37201187260434183501548209472, −4.61024614305815161769641966760, −3.48102537958550228801251168926, −2.89575645816537328892128963103,
0.53405741830788877496693053508, 1.73822863085117964558662835226, 2.43772969753838567112286423909, 3.93126865885455530328782371660, 5.24248267990234540165219500717, 5.55880536221261849997066790054, 7.02993239885298321340728675852, 7.61145889335716296228685849838, 8.653546334423553237225092566378, 9.701192680929505107630909266449