| L(s) = 1 | − 81i·3-s − 473. i·5-s + 574.·7-s − 6.56e3·9-s + 7.27e4i·11-s + 1.48e4i·13-s − 3.83e4·15-s + 1.09e5·17-s − 4.97e5i·19-s − 4.65e4i·21-s + 1.04e5·23-s + 1.72e6·25-s + 5.31e5i·27-s + 7.06e6i·29-s + 2.30e6·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.338i·5-s + 0.0905·7-s − 0.333·9-s + 1.49i·11-s + 0.144i·13-s − 0.195·15-s + 0.319·17-s − 0.875i·19-s − 0.0522i·21-s + 0.0780·23-s + 0.885·25-s + 0.192i·27-s + 1.85i·29-s + 0.448·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.4252903490\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4252903490\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 81iT \) |
| good | 5 | \( 1 + 473. iT - 1.95e6T^{2} \) |
| 7 | \( 1 - 574.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 7.27e4iT - 2.35e9T^{2} \) |
| 13 | \( 1 - 1.48e4iT - 1.06e10T^{2} \) |
| 17 | \( 1 - 1.09e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 4.97e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 - 1.04e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 7.06e6iT - 1.45e13T^{2} \) |
| 31 | \( 1 - 2.30e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 5.20e6iT - 1.29e14T^{2} \) |
| 41 | \( 1 + 1.08e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.04e7iT - 5.02e14T^{2} \) |
| 47 | \( 1 + 3.47e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 9.34e7iT - 3.29e15T^{2} \) |
| 59 | \( 1 - 9.04e7iT - 8.66e15T^{2} \) |
| 61 | \( 1 - 2.00e7iT - 1.16e16T^{2} \) |
| 67 | \( 1 + 1.09e8iT - 2.72e16T^{2} \) |
| 71 | \( 1 + 2.65e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 5.05e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.24e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.72e8iT - 1.86e17T^{2} \) |
| 89 | \( 1 + 2.70e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 4.06e8T + 7.60e17T^{2} \) |
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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
−10.10922579726556324544047458112, −9.165880754331271511428554869554, −8.336749436496172805206605496571, −7.18760147585374173567873930000, −6.74094310350729304726703930762, −5.27357034109771856932543566701, −4.60508872020790287362270796818, −3.19371180620882392297150789291, −2.02241124925216966422200342960, −1.17827954450295474573741768169,
0.079624986441764913443266261091, 1.23457061604569464963924878087, 2.77099000454211224983123177451, 3.49469056370175277415160382302, 4.61248457481747434530646833021, 5.75493028151253811396969200945, 6.42567926296046635348485089126, 7.87642426481831890787017044846, 8.479262349364463640861995254938, 9.596335588910454912354124061788
