| L(s) = 1 | + i·3-s + 4.72·7-s − 9-s + 3.93i·11-s + 3.46i·13-s − 3.51·17-s + 5.44i·19-s + 4.72i·21-s − 7.11·23-s − i·27-s − 3.66i·29-s − 5.23·31-s − 3.93·33-s + 0.414i·37-s − 3.46·39-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.78·7-s − 0.333·9-s + 1.18i·11-s + 0.961i·13-s − 0.852·17-s + 1.24i·19-s + 1.03i·21-s − 1.48·23-s − 0.192i·27-s − 0.681i·29-s − 0.940·31-s − 0.684·33-s + 0.0681i·37-s − 0.555·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.563 - 0.826i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.713780640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.713780640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4.72T + 7T^{2} \) |
| 11 | \( 1 - 3.93iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.51T + 17T^{2} \) |
| 19 | \( 1 - 5.44iT - 19T^{2} \) |
| 23 | \( 1 + 7.11T + 23T^{2} \) |
| 29 | \( 1 + 3.66iT - 29T^{2} \) |
| 31 | \( 1 + 5.23T + 31T^{2} \) |
| 37 | \( 1 - 0.414iT - 37T^{2} \) |
| 41 | \( 1 - 3.00T + 41T^{2} \) |
| 43 | \( 1 - 5.34iT - 43T^{2} \) |
| 47 | \( 1 + 0.925T + 47T^{2} \) |
| 53 | \( 1 - 0.233iT - 53T^{2} \) |
| 59 | \( 1 + 14.3iT - 59T^{2} \) |
| 61 | \( 1 + 0.118iT - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 + 2.19T + 71T^{2} \) |
| 73 | \( 1 - 0.563T + 73T^{2} \) |
| 79 | \( 1 - 10.2T + 79T^{2} \) |
| 83 | \( 1 - 11.3iT - 83T^{2} \) |
| 89 | \( 1 - 8.88T + 89T^{2} \) |
| 97 | \( 1 + 7.27T + 97T^{2} \) |
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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
−9.326688369033811871000154126769, −8.306324766043760808307153081562, −7.915217150386169365750361291533, −7.01503645249672527618011166514, −5.99908975027376595298461795508, −5.10566283054543167341092579238, −4.35021288053816566011586626196, −3.97187843646662349102567771763, −2.18103944734026977771779900726, −1.71148294393198073971123310647,
0.55527370520430703048686584884, 1.73738405749235888007605225903, 2.64524344529039897094731960570, 3.83808702887888216061257819983, 4.87615430162404793393912657828, 5.52149074971882096097091002283, 6.32107135188441243164378296936, 7.40523300402831746966765682328, 7.894862187942756641098064993834, 8.628822998450755543306731099955
