| L(s) = 1 | + (2.5 − 1.65i)3-s − 3.31i·5-s − 2·7-s + (3.5 − 8.29i)9-s + (−5.5 − 8.29i)15-s − 6.63i·17-s − 4·19-s + (−5 + 3.31i)21-s + 3.31i·23-s + 14·25-s + (−4.99 − 26.5i)27-s − 39.7i·29-s + 9·31-s + 6.63i·35-s − 51·37-s + ⋯ |
| L(s) = 1 | + (0.833 − 0.552i)3-s − 0.663i·5-s − 0.285·7-s + (0.388 − 0.921i)9-s + (−0.366 − 0.552i)15-s − 0.390i·17-s − 0.210·19-s + (−0.238 + 0.157i)21-s + 0.144i·23-s + 0.560·25-s + (−0.185 − 0.982i)27-s − 1.37i·29-s + 0.290·31-s + 0.189i·35-s − 1.37·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1452 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.833 + 0.552i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.938928038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.938928038\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.5 + 1.65i)T \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 3.31iT - 25T^{2} \) |
| 7 | \( 1 + 2T + 49T^{2} \) |
| 13 | \( 1 + 169T^{2} \) |
| 17 | \( 1 + 6.63iT - 289T^{2} \) |
| 19 | \( 1 + 4T + 361T^{2} \) |
| 23 | \( 1 - 3.31iT - 529T^{2} \) |
| 29 | \( 1 + 39.7iT - 841T^{2} \) |
| 31 | \( 1 - 9T + 961T^{2} \) |
| 37 | \( 1 + 51T + 1.36e3T^{2} \) |
| 41 | \( 1 + 79.5iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 22T + 1.84e3T^{2} \) |
| 47 | \( 1 - 39.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 66.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 69.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80T + 3.72e3T^{2} \) |
| 67 | \( 1 - T + 4.48e3T^{2} \) |
| 71 | \( 1 + 49.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 108T + 5.32e3T^{2} \) |
| 79 | \( 1 + 122T + 6.24e3T^{2} \) |
| 83 | \( 1 - 86.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 96.1iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 79T + 9.40e3T^{2} \) |
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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
−8.890441173433015700933454699006, −8.324334112349474800676311073443, −7.44791434720883640549828470189, −6.72233839288933335380778908935, −5.77694164563328881709461261407, −4.69232511684925551944274292703, −3.73850491835664958414991910570, −2.74230831927716747855026355429, −1.67625966696058321646923243385, −0.46069352738025453982695313325,
1.64466313615801109297101600716, 2.86237441398598127628461998689, 3.44419860259456825255374431668, 4.49204259966839140526084889890, 5.40199508964753243287663150473, 6.63382690152917199047090495054, 7.17463908653700495114667205596, 8.306729788465449869919132886791, 8.733507965554191785776151735124, 9.774613509938309796461718541903
