| L(s) = 1 | + 27.7i·3-s − 150·5-s + 609. i·7-s − 38.9·9-s − 914. i·11-s + 154·13-s − 4.15e3i·15-s + 7.45e3·17-s + 2.13e3i·19-s − 1.68e4·21-s + 8.48e3i·23-s + 6.87e3·25-s + 1.91e4i·27-s − 1.07e4·29-s − 1.99e3i·31-s + ⋯ |
| L(s) = 1 | + 1.02i·3-s − 1.19·5-s + 1.77i·7-s − 0.0534·9-s − 0.687i·11-s + 0.0700·13-s − 1.23i·15-s + 1.51·17-s + 0.311i·19-s − 1.82·21-s + 0.696i·23-s + 0.440·25-s + 0.971i·27-s − 0.441·29-s − 0.0669i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.500 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.544528 + 0.943151i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.544528 + 0.943151i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 27.7iT - 729T^{2} \) |
| 5 | \( 1 + 150T + 1.56e4T^{2} \) |
| 7 | \( 1 - 609. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 914. iT - 1.77e6T^{2} \) |
| 13 | \( 1 - 154T + 4.82e6T^{2} \) |
| 17 | \( 1 - 7.45e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 2.13e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 8.48e3iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.07e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 1.99e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.13e4T + 2.56e9T^{2} \) |
| 41 | \( 1 - 6.71e4T + 4.75e9T^{2} \) |
| 43 | \( 1 - 7.95e4iT - 6.32e9T^{2} \) |
| 47 | \( 1 + 6.95e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.09e5T + 2.21e10T^{2} \) |
| 59 | \( 1 + 3.05e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 - 3.06e5T + 5.15e10T^{2} \) |
| 67 | \( 1 + 2.20e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 3.72e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.65e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 7.62e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 4.94e5iT - 3.26e11T^{2} \) |
| 89 | \( 1 - 4.71e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 9.10e5T + 8.32e11T^{2} \) |
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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
−18.51424336700464162249999718126, −16.37880943374602554486244483832, −15.61908662502753862843695369547, −14.74744569566762519934132576670, −12.37193444645476470780455908480, −11.31227244744948381628640898034, −9.522930745763892907026271656484, −8.119962954048023734856443129214, −5.45358679998104018047995759692, −3.50829100418603698155215549268,
0.837760838626976873922181219140, 4.05193682631012812589353841347, 7.11660492002165880212406947157, 7.76478140331783440973976323845, 10.35053514129421046638436765303, 11.94493948360992662369585241973, 13.11862411773759558586211064186, 14.47282453352100109102682289760, 16.16595701881246663788934097938, 17.39959232044870421331741843064
