| L(s) = 1 | − 13.7i·2-s − 14.6i·3-s − 125.·4-s + 125. i·5-s − 201.·6-s + 485. i·7-s + 839. i·8-s + 513.·9-s + 1.72e3·10-s − 2.19e3·11-s + 1.83e3i·12-s − 2.48e3·13-s + 6.67e3·14-s + 1.84e3·15-s + 3.54e3·16-s − 3.21e3·17-s + ⋯ |
| L(s) = 1 | − 1.71i·2-s − 0.544i·3-s − 1.95·4-s + 1.00i·5-s − 0.935·6-s + 1.41i·7-s + 1.64i·8-s + 0.703·9-s + 1.72·10-s − 1.64·11-s + 1.06i·12-s − 1.13·13-s + 2.43·14-s + 0.546·15-s + 0.864·16-s − 0.654·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.848 - 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.524957 + 0.150246i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.524957 + 0.150246i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (6.74e4 - 4.20e4i)T \) |
| good | 2 | \( 1 + 13.7iT - 64T^{2} \) |
| 3 | \( 1 + 14.6iT - 729T^{2} \) |
| 5 | \( 1 - 125. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 485. iT - 1.17e5T^{2} \) |
| 11 | \( 1 + 2.19e3T + 1.77e6T^{2} \) |
| 13 | \( 1 + 2.48e3T + 4.82e6T^{2} \) |
| 17 | \( 1 + 3.21e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 828. iT - 4.70e7T^{2} \) |
| 23 | \( 1 - 1.19e4T + 1.48e8T^{2} \) |
| 29 | \( 1 - 3.96e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 4.16e4T + 8.87e8T^{2} \) |
| 37 | \( 1 + 6.79e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.56e4T + 4.75e9T^{2} \) |
| 47 | \( 1 - 9.73e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 7.42e4T + 2.21e10T^{2} \) |
| 59 | \( 1 + 1.59e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 4.54e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 5.92e4T + 9.04e10T^{2} \) |
| 71 | \( 1 - 4.50e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 1.02e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 - 2.66e3T + 2.43e11T^{2} \) |
| 83 | \( 1 + 6.57e5T + 3.26e11T^{2} \) |
| 89 | \( 1 + 6.80e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.25e6T + 8.32e11T^{2} \) |
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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
−14.52005158594166903252852193317, −12.88065514214944358866107693393, −12.57652174809075120326791372844, −11.17340691049427103544929814151, −10.31613290066962044645883177924, −9.036788419776929103467556561617, −7.26250440507474435129563692347, −5.12045775080926097221055994764, −2.90206634267096550531962810278, −2.10949493294047981989542394032,
0.24753120798455589039387095891, 4.45310790883934021642346816437, 5.10686573811960652581143148036, 7.08585163140240941954703200003, 7.911872534636363782375579708783, 9.370688642298319301524758420436, 10.50232920770636812902650048084, 12.95748587527676232023790871643, 13.52800810741256357094594772999, 15.04283944237160121293697243920
