L(s) = 1 | + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (0.5 + 0.866i)23-s + 0.999·28-s + (0.499 − 0.866i)35-s + 0.999·36-s − 43-s + (0.499 + 0.866i)44-s + (0.499 − 0.866i)45-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + (−0.5 − 0.866i)7-s + (−0.5 − 0.866i)9-s + (0.5 − 0.866i)11-s + (−0.499 − 0.866i)16-s + (−1 + 1.73i)17-s + (−0.5 − 0.866i)19-s − 0.999·20-s + (0.5 + 0.866i)23-s + 0.999·28-s + (0.499 − 0.866i)35-s + 0.999·36-s − 43-s + (0.499 + 0.866i)44-s + (0.499 − 0.866i)45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 133 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5751133659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5751133659\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 23 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + T + T^{2} \) |
| 47 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 67 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + T + T^{2} \) |
| 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
−13.51536939115785482243110465220, −12.82004215862498781437345667003, −11.46539040593524054685473501032, −10.59528197589011550738162946023, −9.311234802124883832962565317195, −8.448405537237828755980821676130, −6.94213003987707110179046690510, −6.18638201546974727841309198593, −4.06434645796077638680479019311, −3.09997349348192763214703328180,
2.16502060743839806249047133845, 4.70809796419164559349036462342, 5.42331299502316119487515539401, 6.69334998687290640538404127101, 8.586299636721084614348707081449, 9.239794672657171368592271462543, 10.10117837795661673178713172310, 11.42606310251343268145856726421, 12.61363320452011912066508697198, 13.42880854240873474245589123863