L(s) = 1 | + (−0.5 − 0.866i)3-s + (1 + 1.73i)4-s + (0.5 − 0.866i)5-s + (1 − 1.73i)9-s + (1.5 + 2.59i)11-s + (0.999 − 1.73i)12-s + 5·13-s − 0.999·15-s + (−1.99 + 3.46i)16-s + (−1.5 − 2.59i)17-s + (−1 + 1.73i)19-s + 1.99·20-s + (3 − 5.19i)23-s + (−0.499 − 0.866i)25-s − 5·27-s + ⋯ |
L(s) = 1 | + (−0.288 − 0.499i)3-s + (0.5 + 0.866i)4-s + (0.223 − 0.387i)5-s + (0.333 − 0.577i)9-s + (0.452 + 0.783i)11-s + (0.288 − 0.499i)12-s + 1.38·13-s − 0.258·15-s + (−0.499 + 0.866i)16-s + (−0.363 − 0.630i)17-s + (−0.229 + 0.397i)19-s + 0.447·20-s + (0.625 − 1.08i)23-s + (−0.0999 − 0.173i)25-s − 0.962·27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.126i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.37771 - 0.0874302i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.37771 - 0.0874302i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (-1.5 - 2.59i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 + 5.19i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 3T + 29T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1 - 1.73i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 12T + 41T^{2} \) |
| 43 | \( 1 + 10T + 43T^{2} \) |
| 47 | \( 1 + (4.5 - 7.79i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6 + 10.3i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2 - 3.46i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (1 + 1.73i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-0.5 + 0.866i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 12T + 83T^{2} \) |
| 89 | \( 1 + (-6 + 10.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + T + 97T^{2} \) |
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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
−12.10194804678235806874042783220, −11.45004176964186420458997400265, −10.23014148645109283484995464902, −8.967661317560419698394995409895, −8.173870342928776447521522365671, −6.79828732350708024546049650676, −6.45160428522022824140524885980, −4.65935311714706110465415933365, −3.37297023196331269267382799009, −1.60588733443480794334045228149,
1.64946938923012272135865661838, 3.48941768776716897594935397528, 4.99431691664906587192565670923, 6.03821227235137514661533204502, 6.78827030942083609426156456071, 8.291621576686281165493424515429, 9.440478056710243554149160591124, 10.44868144079610207187074374545, 10.99732400934233756383588807829, 11.65727737368445366670696295834