L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.866 + 0.5i)3-s + (−0.499 − 0.866i)4-s + (−0.696 + 2.12i)5-s + 0.999i·6-s + (1.07 − 0.619i)7-s − 0.999·8-s + (0.499 − 0.866i)9-s + (1.49 + 1.66i)10-s + 2.37·11-s + (0.866 + 0.499i)12-s + (1.12 + 1.94i)13-s − 1.23i·14-s + (−0.459 − 2.18i)15-s + (−0.5 + 0.866i)16-s + (0.499 − 0.865i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.499 + 0.288i)3-s + (−0.249 − 0.433i)4-s + (−0.311 + 0.950i)5-s + 0.408i·6-s + (0.405 − 0.233i)7-s − 0.353·8-s + (0.166 − 0.288i)9-s + (0.471 + 0.526i)10-s + 0.717·11-s + (0.249 + 0.144i)12-s + (0.311 + 0.538i)13-s − 0.330i·14-s + (−0.118 − 0.565i)15-s + (−0.125 + 0.216i)16-s + (0.121 − 0.209i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.539 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.311160627\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.311160627\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.866 - 0.5i)T \) |
| 5 | \( 1 + (0.696 - 2.12i)T \) |
| 37 | \( 1 + (-5.14 + 3.25i)T \) |
good | 7 | \( 1 + (-1.07 + 0.619i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 - 2.37T + 11T^{2} \) |
| 13 | \( 1 + (-1.12 - 1.94i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-0.499 + 0.865i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (7.10 - 4.10i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 1.09T + 23T^{2} \) |
| 29 | \( 1 - 5.72iT - 29T^{2} \) |
| 31 | \( 1 - 3.10iT - 31T^{2} \) |
| 41 | \( 1 + (-4.36 - 7.55i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 - 6.26T + 43T^{2} \) |
| 47 | \( 1 - 6.29iT - 47T^{2} \) |
| 53 | \( 1 + (-10.0 - 5.82i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-1.06 - 0.617i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.52 - 2.61i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.63 + 0.944i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.46 - 4.26i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + 0.123iT - 73T^{2} \) |
| 79 | \( 1 + (13.2 - 7.66i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6.36 + 3.67i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.869 - 0.502i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 5.95T + 97T^{2} \) |
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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
−10.26761544833651039195213403778, −9.384468494472074978741043806638, −8.434741224029335065968102270970, −7.34077043598732338375990234083, −6.42987504511004462225232019511, −5.80865418189900524328652318447, −4.35319891110665422222132005299, −4.02357970503951208776824777641, −2.77607800858589312107388985164, −1.44327579086393163732625489406,
0.58386586634895136535873559092, 2.16592852322758073864358614897, 3.94007487610436145242303885117, 4.54176463958548045159796897973, 5.54454941051254953156818538178, 6.19400308261448392635266232008, 7.15115445244303992391242315479, 8.129731174231044041006910331315, 8.608834096982185235693614598078, 9.487453857622442970744447964023