L(s) = 1 | + (−0.918 − 1.07i)2-s + 1.19·3-s + (−0.313 + 1.97i)4-s + (−2.22 + 0.234i)5-s + (−1.09 − 1.28i)6-s + (−2.11 − 1.58i)7-s + (2.41 − 1.47i)8-s − 1.58·9-s + (2.29 + 2.17i)10-s − 3.65·11-s + (−0.373 + 2.35i)12-s − 5.56i·13-s + (0.239 + 3.73i)14-s + (−2.64 + 0.279i)15-s + (−3.80 − 1.23i)16-s − 0.808·17-s + ⋯ |
L(s) = 1 | + (−0.649 − 0.760i)2-s + 0.687·3-s + (−0.156 + 0.987i)4-s + (−0.994 + 0.105i)5-s + (−0.446 − 0.522i)6-s + (−0.800 − 0.599i)7-s + (0.852 − 0.521i)8-s − 0.527·9-s + (0.725 + 0.688i)10-s − 1.10·11-s + (−0.107 + 0.678i)12-s − 1.54i·13-s + (0.0639 + 0.997i)14-s + (−0.683 + 0.0722i)15-s + (−0.950 − 0.309i)16-s − 0.196·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0117i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0117i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00214824 - 0.364573i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00214824 - 0.364573i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.918 + 1.07i)T \) |
| 5 | \( 1 + (2.22 - 0.234i)T \) |
| 7 | \( 1 + (2.11 + 1.58i)T \) |
good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 11 | \( 1 + 3.65T + 11T^{2} \) |
| 13 | \( 1 + 5.56iT - 13T^{2} \) |
| 17 | \( 1 + 0.808T + 17T^{2} \) |
| 19 | \( 1 + 4.54iT - 19T^{2} \) |
| 23 | \( 1 - 1.75T + 23T^{2} \) |
| 29 | \( 1 - 8.36iT - 29T^{2} \) |
| 31 | \( 1 - 4.73T + 31T^{2} \) |
| 37 | \( 1 + 6.15T + 37T^{2} \) |
| 41 | \( 1 - 7.65iT - 41T^{2} \) |
| 43 | \( 1 - 2.66iT - 43T^{2} \) |
| 47 | \( 1 + 4.79iT - 47T^{2} \) |
| 53 | \( 1 + 2.98T + 53T^{2} \) |
| 59 | \( 1 + 8.92iT - 59T^{2} \) |
| 61 | \( 1 + 4.08T + 61T^{2} \) |
| 67 | \( 1 + 11.7iT - 67T^{2} \) |
| 71 | \( 1 + 8.17iT - 71T^{2} \) |
| 73 | \( 1 - 15.1T + 73T^{2} \) |
| 79 | \( 1 + 2.49iT - 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 + 4.95iT - 89T^{2} \) |
| 97 | \( 1 + 11.7T + 97T^{2} \) |
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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
−11.04568228776966052663535946126, −10.61469080656565439683806934889, −9.513124151114030774256809980927, −8.423121480698989024822757141790, −7.87699132153472659960570728492, −6.91428568633332422054895258108, −4.90582235094899147914747559199, −3.28111985917875609472599222680, −2.95674142093500179135809262767, −0.29620870486625008144351741835,
2.46289025375971538704753878225, 4.04170450598858468444931119454, 5.49000124537857579584719377470, 6.63704953318862227521383589316, 7.70258959110508010664760587369, 8.483496849749157693789793310050, 9.149897893806228134024272452034, 10.14566958143785839777324548294, 11.33487836263995417739175104208, 12.22989852875524086550773678165