L(s) = 1 | + (−0.5 − 0.866i)7-s + (−1.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (1 − 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (1 − 1.73i)79-s − 1.73i·97-s + (−0.5 − 0.866i)109-s + ⋯ |
L(s) = 1 | + (−0.5 − 0.866i)7-s + (−1.5 − 0.866i)19-s + (−0.5 − 0.866i)25-s + (1.5 − 0.866i)31-s + (1 − 1.73i)37-s − 43-s + (−0.499 + 0.866i)49-s + (−1.5 − 0.866i)61-s + (1 + 1.73i)67-s + (−1.5 + 0.866i)73-s + (1 − 1.73i)79-s − 1.73i·97-s + (−0.5 − 0.866i)109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.126 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8885928569\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8885928569\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.5 + 0.866i)T \) |
good | 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 19 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (-1.5 + 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 61 | \( 1 + (1.5 + 0.866i)T + (0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 + T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 97 | \( 1 + 1.73iT - T^{2} \) |
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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
−8.648945698951114820520293425504, −7.965417739014474246010891126742, −7.15165673939126676052198142380, −6.44975179771677594213385084891, −5.85022465390939629681901195272, −4.49439022163802085464188620348, −4.20535541113172442382398115999, −3.02922798616085085357130631764, −2.09677632069388811530758308037, −0.53659147058404712339798999000,
1.57661444517640536435648763308, 2.63365814456375771932375669693, 3.45997084599980878527322066409, 4.48827454076173295228232081585, 5.29546437456800243625744192685, 6.33048902036013736782946845390, 6.49293798176864093981901029232, 7.80051716747258995739023934599, 8.362177901967644800640795167687, 9.054177946641286646910082250703