| L(s) = 1 | − 5.19i·3-s − 42·5-s + 76.2i·7-s − 27·9-s − 20.7i·11-s − 182·13-s + 218. i·15-s − 246·17-s − 117. i·19-s + 396·21-s − 748. i·23-s + 1.13e3·25-s + 140. i·27-s + 78·29-s + 1.47e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.67·5-s + 1.55i·7-s − 0.333·9-s − 0.171i·11-s − 1.07·13-s + 0.969i·15-s − 0.851·17-s − 0.326i·19-s + 0.897·21-s − 1.41i·23-s + 1.82·25-s + 0.192i·27-s + 0.0927·29-s + 1.53i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.0533205 + 0.198994i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0533205 + 0.198994i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 5.19iT \) |
| good | 5 | \( 1 + 42T + 625T^{2} \) |
| 7 | \( 1 - 76.2iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 20.7iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 182T + 2.85e4T^{2} \) |
| 17 | \( 1 + 246T + 8.35e4T^{2} \) |
| 19 | \( 1 + 117. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 748. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 78T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.47e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 530T + 1.87e6T^{2} \) |
| 41 | \( 1 + 918T + 2.82e6T^{2} \) |
| 43 | \( 1 + 852. iT - 3.41e6T^{2} \) |
| 47 | \( 1 - 3.78e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 4.62e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 228. iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 1.34e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.08e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 1.82e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 926T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.39e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 1.19e4iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.15e4T + 6.27e7T^{2} \) |
| 97 | \( 1 + 1.31e4T + 8.85e7T^{2} \) |
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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
−15.37583898284340874073567323191, −14.51395582617085866355526371091, −12.57659196562779654766983422145, −12.11788520647609452952830611026, −11.06364021271221638172635984402, −8.949942189045713755909385347229, −8.040285422325152285462884953920, −6.68722556865995270354279333798, −4.80562169706109947135752656155, −2.74842984673848806790974174259,
0.12304026133366934218400926097, 3.67978856994224562271665549233, 4.59785358008756302741252094972, 7.13957588837042510243482604504, 7.969066972312479192441174093380, 9.737689325801028331617424946321, 10.96605336005405003038884619459, 11.82252072088480578314322528708, 13.31788979148204115279696810690, 14.69813388049315739565035186947
