L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s + 0.999i·8-s + i·11-s + 0.999i·12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + 0.999i·26-s + ⋯ |
L(s) = 1 | + (−0.866 + 0.5i)2-s + (−0.866 + 0.5i)3-s + (0.499 − 0.866i)4-s + (0.499 − 0.866i)6-s + 0.999i·8-s + i·11-s + 0.999i·12-s + (0.5 − 0.866i)13-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (−0.5 − 0.866i)22-s + (0.866 + 0.5i)23-s + (−0.5 − 0.866i)24-s + (0.5 − 0.866i)25-s + 0.999i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3724 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.211 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6377994099\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6377994099\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
good | 3 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 - iT - T^{2} \) |
| 13 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + iT - T^{2} \) |
| 37 | \( 1 - T + T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 73 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
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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
−8.824452165715573277823073973386, −8.085771067604082759010444197639, −7.44781897970534608232472810514, −6.65467996055163970555495581563, −5.87750198315235327173659525833, −5.25216104936584938997412715315, −4.68627806071216564871269862224, −3.40100107124991575497294587643, −2.19010325190831402524207284848, −0.945419366990346898714907974535,
0.77473787396442241495395187797, 1.56615446480715772972578179219, 3.00660564358501641520362332580, 3.52745981939976592603825745047, 4.84683908691249766018604418907, 5.73413210212115335758353735865, 6.52630679242623006411343160220, 7.02128938076699944853280148506, 7.85411226762926728790490735576, 8.623768244908065816972887857590