| L(s) = 1 | + 27i·3-s − 484. i·5-s + 826.·7-s − 729·9-s − 4.55e3i·11-s + 3.07e3i·13-s + 1.30e4·15-s + 2.16e4·17-s + 5.88e4i·19-s + 2.23e4i·21-s + 1.86e4·23-s − 1.56e5·25-s − 1.96e4i·27-s + 2.10e5i·29-s + 9.87e4·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.73i·5-s + 0.911·7-s − 0.333·9-s − 1.03i·11-s + 0.387i·13-s + 1.00·15-s + 1.06·17-s + 1.96i·19-s + 0.526i·21-s + 0.319·23-s − 2.00·25-s − 0.192i·27-s + 1.60i·29-s + 0.595·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.645984122\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.645984122\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27iT \) |
| good | 5 | \( 1 + 484. iT - 7.81e4T^{2} \) |
| 7 | \( 1 - 826.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 4.55e3iT - 1.94e7T^{2} \) |
| 13 | \( 1 - 3.07e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 - 2.16e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 5.88e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 - 1.86e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.10e5iT - 1.72e10T^{2} \) |
| 31 | \( 1 - 9.87e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 5.49e5iT - 9.49e10T^{2} \) |
| 41 | \( 1 - 5.59e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 5.03e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 - 9.70e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.49e6iT - 1.17e12T^{2} \) |
| 59 | \( 1 - 4.15e5iT - 2.48e12T^{2} \) |
| 61 | \( 1 + 2.20e6iT - 3.14e12T^{2} \) |
| 67 | \( 1 + 1.95e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 1.29e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 3.21e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 1.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.28e6iT - 2.71e13T^{2} \) |
| 89 | \( 1 + 3.17e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 2.37e6T + 8.07e13T^{2} \) |
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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
−10.07724840641206734217873247939, −9.081222092469878311512202213924, −8.404411986910037630086690815835, −7.80616472341204184691365975766, −5.92463231818057519637021587641, −5.25347458443284540333348482184, −4.42641608685840575919572572087, −3.40697972626579443963540006249, −1.56091466997500202280361957312, −0.896896471972423486504617760333,
0.70724640334964533342543674038, 2.20713199775570011086119388394, 2.78358064183791528700830156000, 4.21009969990011073080015008926, 5.49968433566552345730659352674, 6.60327955636439850461356684188, 7.38723734199240767309032589007, 7.87734260003044227729962193573, 9.351276381602431828097430585393, 10.31551969798268580244678122803
