| L(s) = 1 | + (−2.04 − 2.04i)3-s + (−0.701 + 0.701i)5-s + i·7-s + 5.33i·9-s + (2.41 − 2.41i)11-s + (−1.96 − 1.96i)13-s + 2.86·15-s − 6.93·17-s + (−1.38 − 1.38i)19-s + (2.04 − 2.04i)21-s + 2.05i·23-s + 4.01i·25-s + (4.76 − 4.76i)27-s + (5.34 + 5.34i)29-s + 5.23·31-s + ⋯ |
| L(s) = 1 | + (−1.17 − 1.17i)3-s + (−0.313 + 0.313i)5-s + 0.377i·7-s + 1.77i·9-s + (0.729 − 0.729i)11-s + (−0.543 − 0.543i)13-s + 0.739·15-s − 1.68·17-s + (−0.318 − 0.318i)19-s + (0.445 − 0.445i)21-s + 0.428i·23-s + 0.803i·25-s + (0.917 − 0.917i)27-s + (0.992 + 0.992i)29-s + 0.940·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7742837004\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7742837004\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + (2.04 + 2.04i)T + 3iT^{2} \) |
| 5 | \( 1 + (0.701 - 0.701i)T - 5iT^{2} \) |
| 11 | \( 1 + (-2.41 + 2.41i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.96 + 1.96i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.93T + 17T^{2} \) |
| 19 | \( 1 + (1.38 + 1.38i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.05iT - 23T^{2} \) |
| 29 | \( 1 + (-5.34 - 5.34i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.23T + 31T^{2} \) |
| 37 | \( 1 + (6.58 - 6.58i)T - 37iT^{2} \) |
| 41 | \( 1 - 0.949iT - 41T^{2} \) |
| 43 | \( 1 + (-5.95 + 5.95i)T - 43iT^{2} \) |
| 47 | \( 1 - 4.64T + 47T^{2} \) |
| 53 | \( 1 + (-7.24 + 7.24i)T - 53iT^{2} \) |
| 59 | \( 1 + (8.58 - 8.58i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.81 - 2.81i)T + 61iT^{2} \) |
| 67 | \( 1 + (9.07 + 9.07i)T + 67iT^{2} \) |
| 71 | \( 1 - 3.60iT - 71T^{2} \) |
| 73 | \( 1 - 6.53iT - 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + (-4.49 - 4.49i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.428iT - 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−9.026082449804762870639575510652, −8.448671555353508940691514927470, −7.36621449444037019674466461433, −6.82002019972894518256163483505, −6.22915179387334884579863034904, −5.39622199603177868186269186638, −4.54700676869268745145836632247, −3.16085054300211177673031460521, −2.02293970671906016740116430959, −0.78273735830871347313984388476,
0.50270154654825922798790171436, 2.26745259399562635357916769005, 3.96564620135661426545027301180, 4.44275952904877152296738119680, 4.84744271211222031409464105478, 6.22058915154418308038477357280, 6.55715228778056583420425175789, 7.64613828837308685447230403963, 8.863553046864342299401325854651, 9.326048451932828895378799426675
