L(s) = 1 | − 5-s + 7-s − 4·11-s + 6·13-s − 6·17-s − 4·23-s + 25-s − 6·29-s − 35-s − 2·37-s − 2·41-s + 4·43-s + 4·47-s + 49-s − 6·53-s + 4·55-s − 12·59-s + 10·61-s − 6·65-s + 4·67-s + 8·71-s − 14·73-s − 4·77-s − 8·79-s + 12·83-s + 6·85-s + 6·89-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 0.377·7-s − 1.20·11-s + 1.66·13-s − 1.45·17-s − 0.834·23-s + 1/5·25-s − 1.11·29-s − 0.169·35-s − 0.328·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s − 0.824·53-s + 0.539·55-s − 1.56·59-s + 1.28·61-s − 0.744·65-s + 0.488·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s − 0.900·79-s + 1.31·83-s + 0.650·85-s + 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.313713566\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.313713566\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−15.68780209943023, −15.31174313361084, −14.63353997735427, −13.76692287751763, −13.62317912135271, −12.87846566658595, −12.55199822968841, −11.56973380629223, −11.24736093036685, −10.75968939531203, −10.31074367223582, −9.418951972591271, −8.773132911996432, −8.376308619251322, −7.774972183520364, −7.238145967255853, −6.405776837758270, −5.898595996380743, −5.208701523400590, −4.460622010167908, −3.905796505591965, −3.210442359273971, −2.295486137691901, −1.635907655983009, −0.4639208283800160,
0.4639208283800160, 1.635907655983009, 2.295486137691901, 3.210442359273971, 3.905796505591965, 4.460622010167908, 5.208701523400590, 5.898595996380743, 6.405776837758270, 7.238145967255853, 7.774972183520364, 8.376308619251322, 8.773132911996432, 9.418951972591271, 10.31074367223582, 10.75968939531203, 11.24736093036685, 11.56973380629223, 12.55199822968841, 12.87846566658595, 13.62317912135271, 13.76692287751763, 14.63353997735427, 15.31174313361084, 15.68780209943023