| L(s) = 1 | − 9i·3-s − 34i·5-s − 240·7-s − 81·9-s + 124i·11-s − 46i·13-s − 306·15-s + 1.95e3·17-s − 1.92e3i·19-s + 2.16e3i·21-s + 2.84e3·23-s + 1.96e3·25-s + 729i·27-s + 8.92e3i·29-s + 4.64e3·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.608i·5-s − 1.85·7-s − 0.333·9-s + 0.308i·11-s − 0.0754i·13-s − 0.351·15-s + 1.63·17-s − 1.22i·19-s + 1.06i·21-s + 1.11·23-s + 0.630·25-s + 0.192i·27-s + 1.97i·29-s + 0.868·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.346306287\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.346306287\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| good | 5 | \( 1 + 34iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 240T + 1.68e4T^{2} \) |
| 11 | \( 1 - 124iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 46iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.95e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.92e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.84e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.92e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 4.64e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.36e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 2.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.13e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 7.00e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.25e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 28iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 6.38e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 + 3.90e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 5.48e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.10e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.66e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 5.61e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.16e5T + 8.58e9T^{2} \) |
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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
−9.205830966753313106370771187425, −8.559600390538474193924553314368, −7.20188586362404889791068556464, −6.84452838479205897290827933294, −5.74677118463012249747854327741, −4.89933720634267778062448848982, −3.40257352582441378151851138634, −2.81528468638764527205549443245, −1.19659117149412787721801474427, −0.36855840676720779825573101285,
0.921973170265991996295108540746, 2.88298712193102379786291314296, 3.21711488267846163890522236695, 4.27383333627080509106621341216, 5.75164611057757585859327778214, 6.21526693846058162364584260093, 7.21099131187461337715254566531, 8.206005299708723249788154091024, 9.324624780421270715640828068591, 10.01128903734225081201664120704
