| L(s) = 1 | − 3.94i·3-s − 0.459·5-s + 4.40i·7-s − 6.58·9-s + 4.56i·11-s − 24.0·13-s + 1.81i·15-s − 19.4·17-s + 0.419i·19-s + 17.4·21-s + 3.76i·23-s − 24.7·25-s − 9.53i·27-s − 5.38·29-s + 40.1i·31-s + ⋯ |
| L(s) = 1 | − 1.31i·3-s − 0.0918·5-s + 0.629i·7-s − 0.731·9-s + 0.415i·11-s − 1.85·13-s + 0.120i·15-s − 1.14·17-s + 0.0220i·19-s + 0.828·21-s + 0.163i·23-s − 0.991·25-s − 0.353i·27-s − 0.185·29-s + 1.29i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 464 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1157382929\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1157382929\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 29 | \( 1 + 5.38T \) |
| good | 3 | \( 1 + 3.94iT - 9T^{2} \) |
| 5 | \( 1 + 0.459T + 25T^{2} \) |
| 7 | \( 1 - 4.40iT - 49T^{2} \) |
| 11 | \( 1 - 4.56iT - 121T^{2} \) |
| 13 | \( 1 + 24.0T + 169T^{2} \) |
| 17 | \( 1 + 19.4T + 289T^{2} \) |
| 19 | \( 1 - 0.419iT - 361T^{2} \) |
| 23 | \( 1 - 3.76iT - 529T^{2} \) |
| 31 | \( 1 - 40.1iT - 961T^{2} \) |
| 37 | \( 1 - 4.71T + 1.36e3T^{2} \) |
| 41 | \( 1 + 19.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 3.64iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 38.3iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 36.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 54.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 19.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 11.7iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 129. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 16.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 94.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 74.1iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 127.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 75.2T + 9.40e3T^{2} \) |
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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
−11.46870917112216582126204905406, −10.17097475222271038872376204039, −9.260349058373212448190295909709, −8.235522646849581981260415044233, −7.31626699283133251375545179302, −6.77995481867770190426178370573, −5.60155467435681173003611810247, −4.49388782881884100149994128977, −2.66166443237625149160999668865, −1.82382005473247197071114963843,
0.04380652457935179408458876191, 2.44866929941052654764901215276, 3.87545995466982188122926298232, 4.54209940198768188272414933225, 5.51340433627426580674298251445, 6.89364055002179592712454788916, 7.81689903200411543766993108680, 9.029650500481842504018777384250, 9.759680890826741042445824533860, 10.38842010475376215993569927036
