L(s) = 1 | + (−1.5 + 0.866i)5-s + (−2.59 − 1.5i)7-s + (2.59 − 4.5i)11-s + (−0.5 − 0.866i)13-s − 3.46i·17-s − 6i·19-s + (−2.59 − 4.5i)23-s + (−1 + 1.73i)25-s + (7.5 + 4.33i)29-s + (−2.59 + 1.5i)31-s + 5.19·35-s − 4·37-s + (−4.5 + 2.59i)41-s + (−2.59 − 1.5i)43-s + (−2.59 + 4.5i)47-s + ⋯ |
L(s) = 1 | + (−0.670 + 0.387i)5-s + (−0.981 − 0.566i)7-s + (0.783 − 1.35i)11-s + (−0.138 − 0.240i)13-s − 0.840i·17-s − 1.37i·19-s + (−0.541 − 0.938i)23-s + (−0.200 + 0.346i)25-s + (1.39 + 0.804i)29-s + (−0.466 + 0.269i)31-s + 0.878·35-s − 0.657·37-s + (−0.702 + 0.405i)41-s + (−0.396 − 0.228i)43-s + (−0.378 + 0.656i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.173 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.531321 - 0.633204i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.531321 - 0.633204i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (1.5 - 0.866i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (2.59 + 1.5i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.59 + 4.5i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (0.5 + 0.866i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 3.46iT - 17T^{2} \) |
| 19 | \( 1 + 6iT - 19T^{2} \) |
| 23 | \( 1 + (2.59 + 4.5i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-7.5 - 4.33i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.59 - 1.5i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 + (4.5 - 2.59i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.59 + 1.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (2.59 - 4.5i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 10.3iT - 53T^{2} \) |
| 59 | \( 1 + (-2.59 - 4.5i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-7.79 + 4.5i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 + (-12.9 - 7.5i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (2.59 - 4.5i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 3.46iT - 89T^{2} \) |
| 97 | \( 1 + (0.5 - 0.866i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.00236751723428702929993568910, −10.06516401614630212499355830310, −9.065481023467991825086194440324, −8.210957469752214655922877825642, −6.91190831286809726056925378783, −6.55107730880931695185568681008, −5.04736710289279896835066056162, −3.69337743861735498078780936030, −2.98602229952351743843644484765, −0.52363058145015849694816103396,
1.88561405254783143931675338137, 3.58536186051464766832966500005, 4.40724668051379413981439624924, 5.82548973559547275081009165802, 6.70689262348291192495058544204, 7.78279468507188299007012152249, 8.688157965834828313886385117844, 9.715881578344310113458260278117, 10.21736546923397834118685118461, 11.85328873377763675528243148773