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 + (3.65 − 0.822i)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 + (0.975 − 0.219i)14-s + (−0.683 − 0.0722i)15-s + (−0.950 + 0.309i)16-s + 0.196·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.270 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.373479 - 0.283054i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.373479 - 0.283054i\) |
\(L(\frac{3}{2})\) |
|
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
−11.25835898651014441054613630906, −10.38327073563453384684820983495, −9.992198284322751612229693438721, −8.798288552272701486504778560446, −7.67417482986994518551211694803, −6.62952236997293920796169767397, −5.72272749487224354902590291119, −5.11223758708300842961236513997, −2.78487683493534377628811868507, −0.45256969043558577848573247443,
1.92450556472681959441494390945, 3.20180678719793615573798593134, 5.03369603558841438016221143563, 6.04341805073640506033174484018, 7.13360926147933164848473220215, 8.697311869190927898467502967069, 9.258214029446342925536882648416, 10.28937655422185420060381886333, 10.92883287579702236825714369475, 12.09336476678506118374260366929