L(s) = 1 | + (−1.33 + 0.461i)2-s + (1.90 + 1.10i)3-s + (1.57 − 1.23i)4-s + (−0.5 − 0.866i)5-s + (−3.05 − 0.591i)6-s + (−0.584 + 2.58i)7-s + (−1.53 + 2.37i)8-s + (0.922 + 1.59i)9-s + (1.06 + 0.926i)10-s + (−2.90 + 5.03i)11-s + (4.35 − 0.620i)12-s + 4.83·13-s + (−0.410 − 3.71i)14-s − 2.20i·15-s + (0.953 − 3.88i)16-s + (3.78 + 2.18i)17-s + ⋯ |
L(s) = 1 | + (−0.945 + 0.326i)2-s + (1.10 + 0.635i)3-s + (0.786 − 0.617i)4-s + (−0.223 − 0.387i)5-s + (−1.24 − 0.241i)6-s + (−0.220 + 0.975i)7-s + (−0.542 + 0.840i)8-s + (0.307 + 0.532i)9-s + (0.337 + 0.293i)10-s + (−0.875 + 1.51i)11-s + (1.25 − 0.179i)12-s + 1.34·13-s + (−0.109 − 0.993i)14-s − 0.568i·15-s + (0.238 − 0.971i)16-s + (0.917 + 0.529i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.145 - 0.989i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.849356 + 0.733910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.849356 + 0.733910i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.33 - 0.461i)T \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (0.584 - 2.58i)T \) |
good | 3 | \( 1 + (-1.90 - 1.10i)T + (1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2.90 - 5.03i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 4.83T + 13T^{2} \) |
| 17 | \( 1 + (-3.78 - 2.18i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.63 + 0.945i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.157 + 0.0911i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.38iT - 29T^{2} \) |
| 31 | \( 1 + (2.03 - 3.53i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.69 + 2.13i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 3.44iT - 41T^{2} \) |
| 43 | \( 1 + 2.10T + 43T^{2} \) |
| 47 | \( 1 + (-0.946 - 1.63i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (8.54 + 4.93i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.35 + 3.66i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.89 + 8.47i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.04 + 10.4i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.80iT - 71T^{2} \) |
| 73 | \( 1 + (-7.34 - 4.24i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-10.5 + 6.09i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 16.9iT - 83T^{2} \) |
| 89 | \( 1 + (5.11 - 2.95i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.2iT - 97T^{2} \) |
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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
−12.01329007859125276862696943611, −10.77189369480286249814679496629, −9.752104506809048724195679478967, −9.255193213975097382900922657036, −8.307547349261119436081917757206, −7.73895123484484312262445810419, −6.22760584491044905467129913385, −5.01344747776955438268955559239, −3.36728972762881173311877552414, −2.04293097289518289546223315070,
1.12395427201998456090608245701, 2.97747495524414601673829046135, 3.51363097036294368873781338733, 6.02587287497276949214846065594, 7.24502014130417746160243611768, 7.947321226983739135699150527127, 8.538465158955042908474696129860, 9.660495952660362403357032885568, 10.78737869468617866257364451060, 11.23508862160880218748199060996