L(s) = 1 | + (−0.5 + 0.866i)2-s + (1 + 1.41i)3-s + (−0.499 − 0.866i)4-s + (0.724 + 1.25i)5-s + (−1.72 + 0.158i)6-s + (0.5 − 0.866i)7-s + 0.999·8-s + (−1.00 + 2.82i)9-s − 1.44·10-s + (−1 + 1.73i)11-s + (0.724 − 1.57i)12-s + (−2.44 − 4.24i)13-s + (0.499 + 0.866i)14-s + (−1.05 + 2.28i)15-s + (−0.5 + 0.866i)16-s + 2·17-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (0.577 + 0.816i)3-s + (−0.249 − 0.433i)4-s + (0.324 + 0.561i)5-s + (−0.704 + 0.0648i)6-s + (0.188 − 0.327i)7-s + 0.353·8-s + (−0.333 + 0.942i)9-s − 0.458·10-s + (−0.301 + 0.522i)11-s + (0.209 − 0.454i)12-s + (−0.679 − 1.17i)13-s + (0.133 + 0.231i)14-s + (−0.271 + 0.588i)15-s + (−0.125 + 0.216i)16-s + 0.485·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.00922 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.755770 + 0.762777i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.755770 + 0.762777i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 + (-1 - 1.41i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
good | 5 | \( 1 + (-0.724 - 1.25i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (2.44 + 4.24i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 2.55T + 19T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.44 + 5.97i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (3 + 5.19i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 11.7T + 37T^{2} \) |
| 41 | \( 1 + (4.89 + 8.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.44 - 5.97i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (4.89 - 8.48i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + (-1 - 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.27 - 5.67i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 - 11.1i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 0.101T + 71T^{2} \) |
| 73 | \( 1 + 6.89T + 73T^{2} \) |
| 79 | \( 1 + (-0.949 + 1.64i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1 - 1.73i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 16.8T + 89T^{2} \) |
| 97 | \( 1 + (1.44 - 2.51i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−14.01333259215675992935018739403, −12.91010140351248478712902324930, −11.20557514430857367363017835591, −10.06587252097080663284932025638, −9.725615955852598893909644741957, −8.148890594196736782438973583706, −7.44229396240704930344692185736, −5.79538136237070114289267443347, −4.56265206948032817000657907568, −2.79404569706707804569198721999,
1.58132965029874222363763505206, 3.12010981632213371051047079961, 5.05631759602413752567637533827, 6.73378771462966882349478729921, 7.998598881879335966217129888521, 8.934326664491490091346343189615, 9.709702493442751698125376658079, 11.28805785214353737047531727260, 12.18434640722822679912361872418, 12.97280171581791195257764470298