L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (−1.73 − i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)37-s + (1.73 + 0.999i)44-s + (−0.999 − 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯ |
L(s) = 1 | + (0.866 + 0.5i)2-s + (0.499 + 0.866i)4-s + 0.999i·8-s + (1.73 − i)11-s + (−0.5 + 0.866i)16-s + 1.99·22-s + (−1.73 − i)23-s + (0.5 + 0.866i)25-s + (−0.866 + 0.499i)32-s + (−1 + 1.73i)37-s + (1.73 + 0.999i)44-s + (−0.999 − 1.73i)46-s + 0.999i·50-s − 0.999·64-s − 2i·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1764 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.605 - 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.981365916\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981365916\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-1.73 + i)T + (0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 17 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 23 | \( 1 + (1.73 + i)T + (0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 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 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 71 | \( 1 + 2iT - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 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
−9.414133468090048691552372361687, −8.604114846485165367377756451998, −8.054065519811579527613742956622, −6.89414551393883860080127986201, −6.39962531471180739800779995331, −5.68903376698362869304282153135, −4.61706481461209074363314089289, −3.82527172320038699897556483198, −3.07882727823879067072400770206, −1.65927051902591981804467826021,
1.45875696382827159344789518132, 2.34089575548995430852041619483, 3.79825535494140145397411078902, 4.10544195877832560964655738112, 5.20454618175549509549389498152, 6.11240860645178291181051337650, 6.78912918638547266293829705291, 7.55310119399822366388242576248, 8.811209351792727814127048820591, 9.587041124781011136057646210977