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 + (−0.499 − 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + 23-s + (0.499 − 0.866i)24-s − 25-s + (0.5 + 0.866i)26-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 + (−0.499 − 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (0.499 − 0.866i)18-s + 23-s + (0.499 − 0.866i)24-s − 25-s + (0.5 + 0.866i)26-s + 0.999·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 276 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.5 - 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7851153577\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7851153577\) |
\(L(1)\) |
|
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 | \( 1 + (0.5 - 0.866i)T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 + T^{2} \) |
| 7 | \( 1 + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T + T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 29 | \( 1 - 1.73iT - T^{2} \) |
| 31 | \( 1 + 1.73iT - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.73iT - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T + T^{2} \) |
| 53 | \( 1 + T^{2} \) |
| 59 | \( 1 + 2T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + T + T^{2} \) |
| 73 | \( 1 + T + T^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 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
−12.48595537714980404353599283399, −11.50553706777584285060768554109, −10.65070831833478680064154049601, −9.357633962448332351498446881115, −8.680969507768690886937833454751, −7.39493287008180899325295428374, −6.22178778357482623339037209118, −5.45020930768995428672893503353, −4.31622591088527698969058230147, −3.32914726616853574232908607697,
1.47366450427510875911822042463, 3.01331157659062557389038820446, 4.53947872496017580081189961752, 5.74849852753773360391225925447, 6.53815911077038375973259754830, 7.945925631455983445365590095950, 9.035389408502922851715141576206, 10.26610519469418080074141594019, 11.16089822397120725876836697606, 11.76729686910859802765950955344