| L(s) = 1 | + 32.4·3-s + 199. i·5-s + 19.6i·7-s + 326.·9-s + 924.·11-s + 1.55e3i·13-s + 6.46e3i·15-s + 5.14e3·17-s + 1.69e3·19-s + 639. i·21-s − 1.92e4i·23-s − 2.40e4·25-s − 1.30e4·27-s − 1.65e4i·29-s − 7.55e3i·31-s + ⋯ |
| L(s) = 1 | + 1.20·3-s + 1.59i·5-s + 0.0573i·7-s + 0.448·9-s + 0.694·11-s + 0.705i·13-s + 1.91i·15-s + 1.04·17-s + 0.247·19-s + 0.0690i·21-s − 1.57i·23-s − 1.53·25-s − 0.663·27-s − 0.680i·29-s − 0.253i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(2.08315 + 1.07737i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.08315 + 1.07737i\) |
| \(L(4)\) |
|
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 | \( 1 - 32.4T + 729T^{2} \) |
| 5 | \( 1 - 199. iT - 1.56e4T^{2} \) |
| 7 | \( 1 - 19.6iT - 1.17e5T^{2} \) |
| 11 | \( 1 - 924.T + 1.77e6T^{2} \) |
| 13 | \( 1 - 1.55e3iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 5.14e3T + 2.41e7T^{2} \) |
| 19 | \( 1 - 1.69e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + 1.92e4iT - 1.48e8T^{2} \) |
| 29 | \( 1 + 1.65e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 7.55e3iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 2.89e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + 5.21e4T + 4.75e9T^{2} \) |
| 43 | \( 1 + 5.89e3T + 6.32e9T^{2} \) |
| 47 | \( 1 + 6.44e4iT - 1.07e10T^{2} \) |
| 53 | \( 1 - 1.97e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 1.42e5T + 4.21e10T^{2} \) |
| 61 | \( 1 + 9.64e4iT - 5.15e10T^{2} \) |
| 67 | \( 1 - 7.52e4T + 9.04e10T^{2} \) |
| 71 | \( 1 + 5.56e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 2.85e5T + 1.51e11T^{2} \) |
| 79 | \( 1 - 3.42e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 - 9.29e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 4.34e5T + 4.96e11T^{2} \) |
| 97 | \( 1 - 6.43e5T + 8.32e11T^{2} \) |
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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
−15.19509297787499299563477711028, −14.43265331425173587733700337463, −13.84143613323513592931858907019, −11.91489971706054742420219687449, −10.50446488912600928161219358262, −9.221536617741077300632421479699, −7.74343218660817683698129424614, −6.46201310166979997261201046370, −3.66940922656360102072760986025, −2.40758719351455111606972062671,
1.30809327864756267662343205828, 3.53722811097475930644493233941, 5.32880736827654231146511780417, 7.77840122948234524373167476073, 8.777647125006386457873678170007, 9.712611688691930086542804384830, 11.90243713205836895111147633620, 13.05122600929239502426416422886, 14.02750636373354738979840162575, 15.26953631941830401562280364735
