| L(s) = 1 | − 1.73i·3-s + 2.23·5-s − 5.46i·7-s − 2.99·9-s + 11.0i·11-s − 10.1·13-s − 3.87i·15-s − 24.4·17-s + 23.7i·19-s − 9.47·21-s + 37.2i·23-s + 5.00·25-s + 5.19i·27-s + 25.7·29-s + 4.83i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.447·5-s − 0.781i·7-s − 0.333·9-s + 1.00i·11-s − 0.778·13-s − 0.258i·15-s − 1.43·17-s + 1.25i·19-s − 0.450·21-s + 1.61i·23-s + 0.200·25-s + 0.192i·27-s + 0.888·29-s + 0.156i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9440006261\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9440006261\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 - 2.23T \) |
| good | 7 | \( 1 + 5.46iT - 49T^{2} \) |
| 11 | \( 1 - 11.0iT - 121T^{2} \) |
| 13 | \( 1 + 10.1T + 169T^{2} \) |
| 17 | \( 1 + 24.4T + 289T^{2} \) |
| 19 | \( 1 - 23.7iT - 361T^{2} \) |
| 23 | \( 1 - 37.2iT - 529T^{2} \) |
| 29 | \( 1 - 25.7T + 841T^{2} \) |
| 31 | \( 1 - 4.83iT - 961T^{2} \) |
| 37 | \( 1 + 35.6T + 1.36e3T^{2} \) |
| 41 | \( 1 + 9.30T + 1.68e3T^{2} \) |
| 43 | \( 1 + 70.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 55.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 82.2T + 3.72e3T^{2} \) |
| 67 | \( 1 - 104. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 76.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 93.5T + 5.32e3T^{2} \) |
| 79 | \( 1 - 49.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 72.3iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 115.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 72.9T + 9.40e3T^{2} \) |
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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
−10.06633703831558245444147099505, −9.291617849485164276926208995679, −8.281866481854084761569143081052, −7.26174674193559446802460984996, −6.92784856104586273477950155059, −5.78108226664285552885839254941, −4.79660105345159371764918403984, −3.78221894755826770551670402629, −2.35635112212887860498338534658, −1.43455627878793867837476283835,
0.28656827504248652817376524546, 2.28411183160345118039927774506, 3.00006790253194522526937148526, 4.50026921925748424253385524662, 5.10715871428701617724316903976, 6.21216743752132326949882696647, 6.82403765561706199780271731152, 8.326936808176312061610871336907, 8.818740797630023708134100268215, 9.519958933654265777604676782906
