L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.580 + 0.211i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.107 + 0.608i)10-s + (−1.23 + 1.04i)13-s + (0.766 + 0.642i)16-s + (−0.280 + 1.59i)17-s + 0.999·18-s + 0.618·20-s + (−0.473 + 0.397i)25-s + (0.809 + 1.40i)26-s + (−0.280 − 1.59i)29-s + (0.766 − 0.642i)32-s + ⋯ |
L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.939 − 0.342i)4-s + (−0.580 + 0.211i)5-s + (−0.5 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.107 + 0.608i)10-s + (−1.23 + 1.04i)13-s + (0.766 + 0.642i)16-s + (−0.280 + 1.59i)17-s + 0.999·18-s + 0.618·20-s + (−0.473 + 0.397i)25-s + (0.809 + 1.40i)26-s + (−0.280 − 1.59i)29-s + (0.766 − 0.642i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.782 - 0.622i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6699186570\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6699186570\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 5 | \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \) |
| 7 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 13 | \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \) |
| 17 | \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
| 23 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 29 | \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 - 0.618T + T^{2} \) |
| 41 | \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \) |
| 43 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 47 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 53 | \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \) |
| 59 | \( 1 + (0.939 + 0.342i)T^{2} \) |
| 61 | \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \) |
| 67 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 71 | \( 1 + (-0.766 + 0.642i)T^{2} \) |
| 73 | \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \) |
| 79 | \( 1 + (-0.173 - 0.984i)T^{2} \) |
| 83 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 89 | \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \) |
| 97 | \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \) |
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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
−9.904725521983053871702308878307, −9.271580853591602288732944228618, −8.143632433862134869418297566134, −7.69883343414819464135310578099, −6.48132483770986526735963228861, −5.43307049901973074638091711580, −4.40646953784592702974863273673, −3.97159968280012428603815074613, −2.57948173831627851045723270462, −1.79808480386251094680648567362,
0.51461945637253604653206781229, 2.86051692980694759961435183125, 3.78495361417254568026315668123, 4.81278174172725645735718563019, 5.39258421306766842729198586702, 6.53666965389014589385047943860, 7.24787034027909224754028033921, 7.81188774167622072915727254168, 8.751441604749329810858438474645, 9.483224284241307686461651207606