| L(s) = 1 | + 1.41i·2-s + (1.84 − 2.36i)3-s − 2.00·4-s + 0.488i·5-s + (3.34 + 2.60i)6-s − 2.82i·8-s + (−2.19 − 8.72i)9-s − 0.690·10-s − 15.1i·11-s + (−3.69 + 4.73i)12-s + 17.3·13-s + (1.15 + 0.900i)15-s + 4.00·16-s − 0.488i·17-s + (12.3 − 3.09i)18-s − 13.0·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.615 − 0.788i)3-s − 0.500·4-s + 0.0976i·5-s + (0.557 + 0.434i)6-s − 0.353i·8-s + (−0.243 − 0.969i)9-s − 0.0690·10-s − 1.37i·11-s + (−0.307 + 0.394i)12-s + 1.33·13-s + (0.0769 + 0.0600i)15-s + 0.250·16-s − 0.0287i·17-s + (0.685 − 0.172i)18-s − 0.687·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.73357 - 0.596190i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.73357 - 0.596190i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-1.84 + 2.36i)T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 0.488iT - 25T^{2} \) |
| 11 | \( 1 + 15.1iT - 121T^{2} \) |
| 13 | \( 1 - 17.3T + 169T^{2} \) |
| 17 | \( 1 + 0.488iT - 289T^{2} \) |
| 19 | \( 1 + 13.0T + 361T^{2} \) |
| 23 | \( 1 + 6.68iT - 529T^{2} \) |
| 29 | \( 1 + 47.3iT - 841T^{2} \) |
| 31 | \( 1 - 28.4T + 961T^{2} \) |
| 37 | \( 1 + T + 1.36e3T^{2} \) |
| 41 | \( 1 + 28.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.14T + 1.84e3T^{2} \) |
| 47 | \( 1 - 73.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 60.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 100. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 34.2T + 3.72e3T^{2} \) |
| 67 | \( 1 + 99.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 82.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 51.7T + 5.32e3T^{2} \) |
| 79 | \( 1 + 66.7T + 6.24e3T^{2} \) |
| 83 | \( 1 - 88.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 58.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 25.0T + 9.40e3T^{2} \) |
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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
−11.52304405626483841114784534598, −10.52561447053211945466709796195, −9.037500878192612438588002235634, −8.507953599246507417745173299277, −7.69306313301148348378270484487, −6.39909797431768254802998202256, −5.96243519900633720155562442775, −4.11706371382500959837952338875, −2.88705829099593044375638349506, −0.921115683599993319628590636427,
1.76622634024379324917685443748, 3.19178264291594637127269547048, 4.26089074644920909922743265941, 5.18314289137193213584127821693, 6.82526076093372467992223405148, 8.255235512952266081766217015138, 8.912676652803953952067608516909, 9.909915845192646606043452997146, 10.58686944486250526365994572269, 11.45642792626462342546752175616
