| L(s) = 1 | − 46.7i·3-s + 36.6i·5-s + (1.27e3 + 2.03e3i)7-s − 2.18e3·9-s + 2.26e4·11-s − 3.61e3i·13-s + 1.71e3·15-s + 4.41e4i·17-s − 1.08e4i·19-s + (9.50e4 − 5.97e4i)21-s + 7.04e4·23-s + 3.89e5·25-s + 1.02e5i·27-s − 4.08e4·29-s + 3.87e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.0586i·5-s + (0.531 + 0.846i)7-s − 0.333·9-s + 1.54·11-s − 0.126i·13-s + 0.0338·15-s + 0.528i·17-s − 0.0830i·19-s + (0.488 − 0.307i)21-s + 0.251·23-s + 0.996·25-s + 0.192i·27-s − 0.0577·29-s + 0.419i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.846 - 0.531i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.654913771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.654913771\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| 7 | \( 1 + (-1.27e3 - 2.03e3i)T \) |
| good | 5 | \( 1 - 36.6iT - 3.90e5T^{2} \) |
| 11 | \( 1 - 2.26e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + 3.61e3iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 4.41e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 + 1.08e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 7.04e4T + 7.83e10T^{2} \) |
| 29 | \( 1 + 4.08e4T + 5.00e11T^{2} \) |
| 31 | \( 1 - 3.87e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.75e5T + 3.51e12T^{2} \) |
| 41 | \( 1 + 1.44e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.35e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.28e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 3.39e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.77e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.03e7iT - 1.91e14T^{2} \) |
| 67 | \( 1 + 1.12e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 1.27e7T + 6.45e14T^{2} \) |
| 73 | \( 1 + 1.88e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.08e7T + 1.51e15T^{2} \) |
| 83 | \( 1 + 3.35e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 2.39e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 4.47e7iT - 7.83e15T^{2} \) |
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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
−10.30818891544038564709986106625, −8.971818287354618866719904412154, −8.612103164878589272385177789767, −7.33706972399371670909989758905, −6.45589902911959531826347029456, −5.55440608371739911960634961047, −4.34249595923333990624299176035, −3.05957802115988189065001973042, −1.84968897528386448327123321494, −1.00294127090219097620530261705,
0.61896322356235210179843528619, 1.62609563848955459165030018155, 3.22326994452938736465511731711, 4.19926074999356507747863254601, 4.94062405991194687260038154684, 6.32564658704960487449959284745, 7.18965651083741098780592880212, 8.339556729504486720477110265158, 9.260154553493258887171641092612, 10.01941461402495070744531890449
