L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + 17-s + (0.499 + 0.866i)18-s + 2·19-s + (−0.499 − 0.866i)22-s + (−0.499 − 0.866i)24-s − 0.999·27-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + 0.999·6-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (0.5 − 0.866i)11-s + (0.499 − 0.866i)12-s + (−0.5 + 0.866i)16-s + 17-s + (0.499 + 0.866i)18-s + 2·19-s + (−0.499 − 0.866i)22-s + (−0.499 − 0.866i)24-s − 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.681569979\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.681569979\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 17 | \( 1 - T + T^{2} \) |
| 19 | \( 1 - 2T + T^{2} \) |
| 23 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 2T + T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + T + T^{2} \) |
| 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
−9.625680110464385798226153276670, −8.829115645374614438163385918510, −8.122281476155593061212566328679, −6.96916159648424128705777112283, −5.53738507166212527556685598674, −5.39525608326456279268509654776, −4.12925643737480546430311282478, −3.42510382577416124173511658769, −2.80555211681645716994956358541, −1.32656648343528906529127900678,
1.43876718605282742789361477591, 2.94069395113731444155815619756, 3.59544207078539963036210579025, 4.81099938687918431875492147447, 5.63338561099498685332799190520, 6.52532524697068355893826548040, 7.22642292379932214219448988125, 7.75427263841609384953768013927, 8.465221021293952264873400573034, 9.471096139337727433943782610534