| L(s) = 1 | − 22.6i·2-s − 512.·4-s + 1.39e3i·5-s + 5.67e3·7-s + 1.15e4i·8-s + 3.16e4·10-s − 7.48e4i·11-s − 6.37e5·13-s − 1.28e5i·14-s + 2.62e5·16-s + 2.44e6i·17-s + 3.94e6·19-s − 7.15e5i·20-s − 1.69e6·22-s − 5.51e6i·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.500·4-s + 0.447i·5-s + 0.337·7-s + 0.353i·8-s + 0.316·10-s − 0.464i·11-s − 1.71·13-s − 0.238i·14-s + 0.250·16-s + 1.72i·17-s + 1.59·19-s − 0.223i·20-s − 0.328·22-s − 0.857i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.577 + 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.593525 - 1.14660i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.593525 - 1.14660i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 22.6iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 1.39e3iT \) |
| good | 7 | \( 1 - 5.67e3T + 2.82e8T^{2} \) |
| 11 | \( 1 + 7.48e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 6.37e5T + 1.37e11T^{2} \) |
| 17 | \( 1 - 2.44e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 3.94e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 5.51e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 2.28e7iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 3.50e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 1.06e8T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.39e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 - 2.63e8T + 2.16e16T^{2} \) |
| 47 | \( 1 + 2.27e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 1.76e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 + 5.62e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 + 1.29e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 5.86e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 3.03e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 + 8.55e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + 4.17e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 5.60e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 7.16e9iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 6.54e9T + 7.37e19T^{2} \) |
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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
−11.76449376557215734606465374052, −10.57536159162549314157270208445, −9.825948476695803572433528358090, −8.455977830205283424354891587777, −7.32144820558987897084680186845, −5.76072814469998094041579823278, −4.44404827799136160853249227135, −3.08032623486128419640874632972, −1.91689609024495689977070477859, −0.37323068550112758061784272582,
1.07976130413718055535239662653, 2.85723763933273365623602595867, 4.73258218398514206626720889949, 5.29443587398585447708246959026, 7.09927763919752142642467238719, 7.70028829331285479878119530800, 9.238566431974039227054937660658, 9.865915408063098472993361084827, 11.63629890449104358285209578584, 12.42037259849582955632198559180
