L(s) = 1 | + 2·5-s − 4·7-s − 11-s + 6·13-s − 6·17-s + 8·19-s − 25-s + 6·29-s − 8·35-s + 6·37-s + 10·41-s + 8·43-s + 9·49-s − 6·53-s − 2·55-s + 4·59-s − 2·61-s + 12·65-s + 12·67-s − 8·71-s + 2·73-s + 4·77-s + 4·79-s − 12·83-s − 12·85-s + 6·89-s − 24·91-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.51·7-s − 0.301·11-s + 1.66·13-s − 1.45·17-s + 1.83·19-s − 1/5·25-s + 1.11·29-s − 1.35·35-s + 0.986·37-s + 1.56·41-s + 1.21·43-s + 9/7·49-s − 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.256·61-s + 1.48·65-s + 1.46·67-s − 0.949·71-s + 0.234·73-s + 0.455·77-s + 0.450·79-s − 1.31·83-s − 1.30·85-s + 0.635·89-s − 2.51·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.776127903\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.776127903\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−9.402612326378998416643459368370, −8.919906421356101286450384033508, −7.81845995144891541405433174054, −6.76153137764114692580153019196, −6.15707753854340364126761944921, −5.62329098122753327257506148467, −4.29789560350206921563462558010, −3.29333770957049475081943753861, −2.45750776683153373018171759402, −0.959729037876911481426237589919,
0.959729037876911481426237589919, 2.45750776683153373018171759402, 3.29333770957049475081943753861, 4.29789560350206921563462558010, 5.62329098122753327257506148467, 6.15707753854340364126761944921, 6.76153137764114692580153019196, 7.81845995144891541405433174054, 8.919906421356101286450384033508, 9.402612326378998416643459368370