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
−11.23508862160880218748199060996, −10.78737869468617866257364451060, −9.660495952660362403357032885568, −8.538465158955042908474696129860, −7.947321226983739135699150527127, −7.24502014130417746160243611768, −6.02587287497276949214846065594, −3.51363097036294368873781338733, −2.97747495524414601673829046135, −1.12395427201998456090608245701,
2.04293097289518289546223315070, 3.36728972762881173311877552414, 5.01344747776955438268955559239, 6.22760584491044905467129913385, 7.73895123484484312262445810419, 8.307547349261119436081917757206, 9.255193213975097382900922657036, 9.752104506809048724195679478967, 10.77189369480286249814679496629, 12.01329007859125276862696943611