| L(s) = 1 | − 4.99i·7-s + 3.99·11-s − 1.54i·13-s − 6.99i·17-s + 2.25·19-s − 7.79i·23-s − 6.16·29-s − 0.543·31-s − 6.25i·37-s + 0.195·41-s + 0.0863i·43-s − 3.82i·47-s − 17.9·49-s − 4.19i·53-s − 7.02·59-s + ⋯ |
| L(s) = 1 | − 1.88i·7-s + 1.20·11-s − 0.427i·13-s − 1.69i·17-s + 0.517·19-s − 1.62i·23-s − 1.14·29-s − 0.0975·31-s − 1.02i·37-s + 0.0305·41-s + 0.0131i·43-s − 0.558i·47-s − 2.56·49-s − 0.575i·53-s − 0.914·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.855687767\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.855687767\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4.99iT - 7T^{2} \) |
| 11 | \( 1 - 3.99T + 11T^{2} \) |
| 13 | \( 1 + 1.54iT - 13T^{2} \) |
| 17 | \( 1 + 6.99iT - 17T^{2} \) |
| 19 | \( 1 - 2.25T + 19T^{2} \) |
| 23 | \( 1 + 7.79iT - 23T^{2} \) |
| 29 | \( 1 + 6.16T + 29T^{2} \) |
| 31 | \( 1 + 0.543T + 31T^{2} \) |
| 37 | \( 1 + 6.25iT - 37T^{2} \) |
| 41 | \( 1 - 0.195T + 41T^{2} \) |
| 43 | \( 1 - 0.0863iT - 43T^{2} \) |
| 47 | \( 1 + 3.82iT - 47T^{2} \) |
| 53 | \( 1 + 4.19iT - 53T^{2} \) |
| 59 | \( 1 + 7.02T + 59T^{2} \) |
| 61 | \( 1 + 2.90T + 61T^{2} \) |
| 67 | \( 1 - 8.96iT - 67T^{2} \) |
| 71 | \( 1 - 8.79T + 71T^{2} \) |
| 73 | \( 1 - 2.28iT - 73T^{2} \) |
| 79 | \( 1 - 12.6T + 79T^{2} \) |
| 83 | \( 1 - 13.9iT - 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 9.33iT - 97T^{2} \) |
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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
−7.43660915960963869456081891629, −6.90157173828704619768794817833, −6.41677053354211109994674786736, −5.30939564653570794967267055809, −4.62575835670120980748156458286, −3.91069484470668843582639048841, −3.39285023646489006406492090150, −2.28149277992284773221261973784, −1.04566638966175294150753829891, −0.47980898893586063898836111896,
1.57031658977482468064438691083, 1.89960383892651933481354902746, 3.14806475096557762231963977623, 3.68925821320264957667794484304, 4.66989565734745706627877726336, 5.50317996708208117255428104911, 6.06962668426376262102940551174, 6.46823021988515165808848823987, 7.54692511394726178908803278493, 8.153205959821156757107139313575
