L(s) = 1 | + (0.721 + 2.91i)3-s + 2.23i·5-s + 1.03·7-s + (−7.95 + 4.20i)9-s − 19.5i·11-s + 16.3·13-s + (−6.51 + 1.61i)15-s − 27.1i·17-s + 24.3·19-s + (0.747 + 3.01i)21-s + 19.9i·23-s − 5.00·25-s + (−17.9 − 20.1i)27-s − 50.0i·29-s + 32.9·31-s + ⋯ |
L(s) = 1 | + (0.240 + 0.970i)3-s + 0.447i·5-s + 0.148·7-s + (−0.884 + 0.466i)9-s − 1.77i·11-s + 1.25·13-s + (−0.434 + 0.107i)15-s − 1.59i·17-s + 1.28·19-s + (0.0356 + 0.143i)21-s + 0.866i·23-s − 0.200·25-s + (−0.665 − 0.746i)27-s − 1.72i·29-s + 1.06·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.240i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.160650952\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.160650952\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.721 - 2.91i)T \) |
| 5 | \( 1 - 2.23iT \) |
good | 7 | \( 1 - 1.03T + 49T^{2} \) |
| 11 | \( 1 + 19.5iT - 121T^{2} \) |
| 13 | \( 1 - 16.3T + 169T^{2} \) |
| 17 | \( 1 + 27.1iT - 289T^{2} \) |
| 19 | \( 1 - 24.3T + 361T^{2} \) |
| 23 | \( 1 - 19.9iT - 529T^{2} \) |
| 29 | \( 1 + 50.0iT - 841T^{2} \) |
| 31 | \( 1 - 32.9T + 961T^{2} \) |
| 37 | \( 1 - 0.942T + 1.36e3T^{2} \) |
| 41 | \( 1 + 11.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 53.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 68.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 48.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 14.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 34.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 9.54T + 4.48e3T^{2} \) |
| 71 | \( 1 + 10.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 14.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 81.0T + 6.24e3T^{2} \) |
| 83 | \( 1 + 58.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 37.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 96.7T + 9.40e3T^{2} \) |
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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
−9.725634456668123131256273688027, −9.188300863667354485404155177931, −8.265678833775960041266769568879, −7.57030780143599984126922012026, −6.11505608783874874247540093193, −5.63695378790813221034528519764, −4.45258468860424158850698385204, −3.36143257376020253680761690336, −2.84506199388205164288765140542, −0.802246662169803549170736953755,
1.17052964293948886700981120876, 1.94671736494811158129128569247, 3.35526508900465196450100687599, 4.48127754311578316188405161317, 5.57583288867224060103740851406, 6.53151758824243328255238615113, 7.25164913959088138099728817397, 8.199669457003110954083913222144, 8.717984252354796666067263982640, 9.720332191854287339735314374159