| L(s) = 1 | − 0.364i·3-s + 2.86·9-s + 1.36·11-s − 2.63i·13-s − 2.23i·17-s − 6.23·19-s − 6.59i·23-s − 2.13i·27-s − 5.50·29-s − 4.50·31-s − 0.497i·33-s − 2.09i·37-s − 0.960·39-s − 9.32·41-s + 1.86i·43-s + ⋯ |
| L(s) = 1 | − 0.210i·3-s + 0.955·9-s + 0.411·11-s − 0.730i·13-s − 0.541i·17-s − 1.42·19-s − 1.37i·23-s − 0.411i·27-s − 1.02·29-s − 0.808·31-s − 0.0865i·33-s − 0.344i·37-s − 0.153·39-s − 1.45·41-s + 0.284i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9055829469\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9055829469\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 0.364iT - 3T^{2} \) |
| 11 | \( 1 - 1.36T + 11T^{2} \) |
| 13 | \( 1 + 2.63iT - 13T^{2} \) |
| 17 | \( 1 + 2.23iT - 17T^{2} \) |
| 19 | \( 1 + 6.23T + 19T^{2} \) |
| 23 | \( 1 + 6.59iT - 23T^{2} \) |
| 29 | \( 1 + 5.50T + 29T^{2} \) |
| 31 | \( 1 + 4.50T + 31T^{2} \) |
| 37 | \( 1 + 2.09iT - 37T^{2} \) |
| 41 | \( 1 + 9.32T + 41T^{2} \) |
| 43 | \( 1 - 1.86iT - 43T^{2} \) |
| 47 | \( 1 - 6.86iT - 47T^{2} \) |
| 53 | \( 1 - 10.1iT - 53T^{2} \) |
| 59 | \( 1 + 1.63T + 59T^{2} \) |
| 61 | \( 1 + 0.0394T + 61T^{2} \) |
| 67 | \( 1 + 6.72iT - 67T^{2} \) |
| 71 | \( 1 + 6.27T + 71T^{2} \) |
| 73 | \( 1 + 4iT - 73T^{2} \) |
| 79 | \( 1 + 5.32T + 79T^{2} \) |
| 83 | \( 1 + 14.7iT - 83T^{2} \) |
| 89 | \( 1 - 0.867T + 89T^{2} \) |
| 97 | \( 1 - 16.0iT - 97T^{2} \) |
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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
−7.82780741335785530611356955415, −7.28863617364427412630805466518, −6.50592462913593817631077689515, −5.94526589714988075861175181809, −4.86098930329164254082623356595, −4.29861548394119704624792369404, −3.41449209312322340114034187065, −2.37127712104366071053375726362, −1.49365673382309313282098500139, −0.23110888732507119920268569167,
1.51740990352435965824076084425, 2.06499755665298772220029488291, 3.61359032381666833282292292647, 3.92468756194185530569542791192, 4.83979928389079672123582482542, 5.62938475986696776416691133332, 6.56633364522583466003589453989, 7.02431003981589570620445273433, 7.80571569789522161006945976531, 8.708812324856149729748231377491
