L(s) = 1 | + (0.930 + 1.46i)3-s + 1.36·5-s + (2.64 + 0.0810i)7-s + (−1.26 + 2.71i)9-s − 1.20i·11-s + (0.639 + 0.369i)13-s + (1.27 + 1.99i)15-s + (−0.693 + 1.20i)17-s + (2.81 − 1.62i)19-s + (2.34 + 3.93i)21-s − 3.81i·23-s − 3.12·25-s + (−5.15 + 0.677i)27-s + (−3.50 + 2.02i)29-s + (1.02 − 0.594i)31-s + ⋯ |
L(s) = 1 | + (0.537 + 0.843i)3-s + 0.611·5-s + (0.999 + 0.0306i)7-s + (−0.422 + 0.906i)9-s − 0.362i·11-s + (0.177 + 0.102i)13-s + (0.328 + 0.516i)15-s + (−0.168 + 0.291i)17-s + (0.646 − 0.373i)19-s + (0.511 + 0.859i)21-s − 0.794i·23-s − 0.625·25-s + (−0.991 + 0.130i)27-s + (−0.649 + 0.375i)29-s + (0.184 − 0.106i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.636 - 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.82072 + 0.858789i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.82072 + 0.858789i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.930 - 1.46i)T \) |
| 7 | \( 1 + (-2.64 - 0.0810i)T \) |
good | 5 | \( 1 - 1.36T + 5T^{2} \) |
| 11 | \( 1 + 1.20iT - 11T^{2} \) |
| 13 | \( 1 + (-0.639 - 0.369i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.693 - 1.20i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.81 + 1.62i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + 3.81iT - 23T^{2} \) |
| 29 | \( 1 + (3.50 - 2.02i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.02 + 0.594i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-5.10 - 8.84i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.670 - 1.16i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.490 + 0.848i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.63 - 2.83i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (5.77 + 3.33i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.73 + 11.6i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.36 + 2.51i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (2.19 + 3.80i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.84iT - 71T^{2} \) |
| 73 | \( 1 + (-10.2 - 5.94i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.34 + 10.9i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.14 + 5.44i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (6.05 + 10.4i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.9 + 6.33i)T + (48.5 - 84.0i)T^{2} \) |
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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
−11.01370284431632641611693169719, −10.07314280824633458203059907192, −9.315617080371298703636650123900, −8.435464872777616963524841782615, −7.73830223339326057871443917185, −6.27732723046593952771214729639, −5.21564071750493894929961680283, −4.41344366601713667947764916299, −3.11608419032397303821916017255, −1.82982749506599279963717509189,
1.41133993255106644676072905741, 2.41380831694203354161354467181, 3.87918573112802128413467573410, 5.30724739494807464717574898782, 6.17136645496828338183496825585, 7.44042204022834396938848372586, 7.87003832202152417858644194788, 9.035879983315932807004391999127, 9.678389213994835297610288979922, 10.91691029099805003132865304275