| L(s) = 1 | − 0.528i·3-s + 5-s + (−2.17 + 1.50i)7-s + 2.72·9-s − 3.04·11-s − 4.75·13-s − 0.528i·15-s + 6.78i·17-s + 0.584i·19-s + (0.796 + 1.14i)21-s + 5.80i·23-s + 25-s − 3.02i·27-s − 0.185i·29-s − 3.12·31-s + ⋯ |
| L(s) = 1 | − 0.304i·3-s + 0.447·5-s + (−0.821 + 0.569i)7-s + 0.907·9-s − 0.918·11-s − 1.31·13-s − 0.136i·15-s + 1.64i·17-s + 0.134i·19-s + (0.173 + 0.250i)21-s + 1.21i·23-s + 0.200·25-s − 0.581i·27-s − 0.0344i·29-s − 0.561·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.263 - 0.964i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9500746769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9500746769\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + (2.17 - 1.50i)T \) |
| good | 3 | \( 1 + 0.528iT - 3T^{2} \) |
| 11 | \( 1 + 3.04T + 11T^{2} \) |
| 13 | \( 1 + 4.75T + 13T^{2} \) |
| 17 | \( 1 - 6.78iT - 17T^{2} \) |
| 19 | \( 1 - 0.584iT - 19T^{2} \) |
| 23 | \( 1 - 5.80iT - 23T^{2} \) |
| 29 | \( 1 + 0.185iT - 29T^{2} \) |
| 31 | \( 1 + 3.12T + 31T^{2} \) |
| 37 | \( 1 - 7.04iT - 37T^{2} \) |
| 41 | \( 1 + 3.83iT - 41T^{2} \) |
| 43 | \( 1 - 1.43T + 43T^{2} \) |
| 47 | \( 1 + 2.95T + 47T^{2} \) |
| 53 | \( 1 + 0.535iT - 53T^{2} \) |
| 59 | \( 1 - 1.52iT - 59T^{2} \) |
| 61 | \( 1 + 13.2T + 61T^{2} \) |
| 67 | \( 1 - 9.13T + 67T^{2} \) |
| 71 | \( 1 - 9.68iT - 71T^{2} \) |
| 73 | \( 1 - 12.4iT - 73T^{2} \) |
| 79 | \( 1 - 2.81iT - 79T^{2} \) |
| 83 | \( 1 - 13.4iT - 83T^{2} \) |
| 89 | \( 1 + 3.10iT - 89T^{2} \) |
| 97 | \( 1 + 13.5iT - 97T^{2} \) |
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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
−9.972149452149855804557686578262, −9.487472235536006494344611734425, −8.371020169910624364007021955499, −7.53554831199302118214399637095, −6.75644280713673005407664415932, −5.85246399565123798485267790229, −5.08884711078767525429008228010, −3.85559974897970191203942672705, −2.67478280505096565442941077305, −1.68928600649497031884803940233,
0.39270123350847926758121241318, 2.28365773572658033295117739797, 3.20215849517687733732050740685, 4.55374899338979603920708243765, 5.06307744208129248323951130550, 6.31424971875586880319753857310, 7.22719482234411543159250650846, 7.63193586953803082738787005219, 9.154766737364159036934324659459, 9.615711917319350225498973255763
