L(s) = 1 | − i·3-s + (0.5 − 0.866i)5-s − 9-s + (−0.866 − 0.5i)11-s − 13-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·27-s + (−0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + i·39-s − 2i·43-s + ⋯ |
L(s) = 1 | − i·3-s + (0.5 − 0.866i)5-s − 9-s + (−0.866 − 0.5i)11-s − 13-s + (−0.866 − 0.5i)15-s + (−0.5 + 0.866i)17-s + (−0.866 − 0.5i)19-s + i·27-s + (−0.5 + 0.866i)29-s + (−0.866 − 0.5i)31-s + (−0.5 + 0.866i)33-s + (−0.5 − 0.866i)37-s + i·39-s − 2i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3744 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5718186051\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5718186051\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| 13 | \( 1 + T \) |
good | 5 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 + 2iT - T^{2} \) |
| 47 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 - 2T + T^{2} \) |
| 79 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + T^{2} \) |
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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
−8.357927662309441000338273143997, −7.48783257382055484973896580869, −6.94071677654204747755156958540, −5.93030086220552215748286187389, −5.43900553299774211637137310957, −4.69362597518006149450560628997, −3.47961866012247890660246005782, −2.32945083273958573742644884867, −1.76877654736232833105143235932, −0.28781537573003314219619822609,
2.21922387356864598325314033991, 2.69376347872933197751907771488, 3.67743343580477973851153233018, 4.72160685213825204841323004927, 5.14034222876406892175774642307, 6.13795091711257480392306527663, 6.79573873328545559449639996598, 7.68278236419835727990859790227, 8.360845529085327429964977276878, 9.471691554670746466429081606600