| L(s) = 1 | + 2.14i·3-s − 1.41·5-s − 7.84i·7-s + 4.37·9-s + 20.2i·11-s + 21.9·13-s − 3.04i·15-s + 4.12·17-s + 8.96i·19-s + 16.8·21-s + 33.7i·23-s − 23.0·25-s + 28.7i·27-s − 36.1·29-s − 0.236i·31-s + ⋯ |
| L(s) = 1 | + 0.716i·3-s − 0.282·5-s − 1.12i·7-s + 0.486·9-s + 1.84i·11-s + 1.68·13-s − 0.202i·15-s + 0.242·17-s + 0.471i·19-s + 0.803·21-s + 1.46i·23-s − 0.920·25-s + 1.06i·27-s − 1.24·29-s − 0.00762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 272 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.41558 + 0.817289i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.41558 + 0.817289i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 17 | \( 1 - 4.12T \) |
| good | 3 | \( 1 - 2.14iT - 9T^{2} \) |
| 5 | \( 1 + 1.41T + 25T^{2} \) |
| 7 | \( 1 + 7.84iT - 49T^{2} \) |
| 11 | \( 1 - 20.2iT - 121T^{2} \) |
| 13 | \( 1 - 21.9T + 169T^{2} \) |
| 19 | \( 1 - 8.96iT - 361T^{2} \) |
| 23 | \( 1 - 33.7iT - 529T^{2} \) |
| 29 | \( 1 + 36.1T + 841T^{2} \) |
| 31 | \( 1 + 0.236iT - 961T^{2} \) |
| 37 | \( 1 - 71.2T + 1.36e3T^{2} \) |
| 41 | \( 1 - 67.1T + 1.68e3T^{2} \) |
| 43 | \( 1 + 62.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 33.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 6.21iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 8.71T + 3.72e3T^{2} \) |
| 67 | \( 1 - 75.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 30.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 51.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 10.3iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 105. iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 132.T + 7.92e3T^{2} \) |
| 97 | \( 1 + 53.4T + 9.40e3T^{2} \) |
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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
−11.66702631324833373639100520221, −10.75799797185823041433222795661, −9.967144938803069107258582316904, −9.293396612870134276644064482329, −7.72308753221785140141782880803, −7.17784513492217602270432181530, −5.68329548881696000502543267225, −4.14646627133206213879950530929, −3.89113368698376323391416211016, −1.51945459438942793921458612564,
0.976114379531936843853843424703, 2.68886234377548708114220568952, 4.05397557389841085933348000687, 5.89511550093353861895599267246, 6.23824088581963782817817648902, 7.82905751768173637701876776112, 8.490336068800507845240517191800, 9.386663855450418520920558572438, 11.01735679391154355824976051742, 11.38519891113995505670525461041
