L(s) = 1 | + (2.44 + 1.41i)3-s + (−1.04 − 1.81i)7-s + (2.49 + 4.32i)9-s + (1.5 + 0.866i)11-s + (−0.331 + 3.59i)13-s + (3.14 − 1.81i)17-s + (−0.926 + 0.534i)19-s − 5.92i·21-s + (6.77 + 3.90i)23-s + 5.62i·27-s + (−0.263 + 0.456i)29-s + 5.84i·31-s + (2.44 + 4.24i)33-s + (−4.87 + 8.44i)37-s + (−5.88 + 8.32i)39-s + ⋯ |
L(s) = 1 | + (1.41 + 0.816i)3-s + (−0.395 − 0.685i)7-s + (0.831 + 1.44i)9-s + (0.452 + 0.261i)11-s + (−0.0918 + 0.995i)13-s + (0.762 − 0.439i)17-s + (−0.212 + 0.122i)19-s − 1.29i·21-s + (1.41 + 0.815i)23-s + 1.08i·27-s + (−0.0489 + 0.0847i)29-s + 1.04i·31-s + (0.426 + 0.738i)33-s + (−0.801 + 1.38i)37-s + (−0.942 + 1.33i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.516 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.714834393\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.714834393\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + (0.331 - 3.59i)T \) |
good | 3 | \( 1 + (-2.44 - 1.41i)T + (1.5 + 2.59i)T^{2} \) |
| 7 | \( 1 + (1.04 + 1.81i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 0.866i)T + (5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.14 + 1.81i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.926 - 0.534i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.77 - 3.90i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.263 - 0.456i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 5.84iT - 31T^{2} \) |
| 37 | \( 1 + (4.87 - 8.44i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.69 + 2.13i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-8.09 + 4.67i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + 12.5iT - 53T^{2} \) |
| 59 | \( 1 + (-1.21 + 0.701i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.55 - 9.62i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.41 + 9.38i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (12.2 - 7.08i)T + (35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 2.64T + 73T^{2} \) |
| 79 | \( 1 + 13.5T + 79T^{2} \) |
| 83 | \( 1 + 15.7T + 83T^{2} \) |
| 89 | \( 1 + (-4.78 - 2.76i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (7.59 + 13.1i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−9.824303568696996131457909806228, −8.927964087177345271943800579117, −8.501890461454698207538664042393, −7.22692997394872676028634125286, −6.93945162705227488756018467263, −5.33566533600925463933327075948, −4.37535909184439234691469129446, −3.62929623649893108948235571052, −2.90605863736703013975558169121, −1.55313872269352465061618163427,
1.07826631802739986086674500724, 2.45759816110481742639502518202, 3.02940478931307364545516686689, 4.02467202922743716353263963720, 5.50128502963592439403919233264, 6.32295537995586847624337166770, 7.31848383078671951689203882503, 7.898108612358166600184350254309, 8.825455208045696854835000111897, 9.102261772454638582154733805327